Contact processes between deformable bodies abound in industry and everyday life and, for this reason, considerable efforts have been made in their modelling and analysis. Owing to their inherent complexity, contact phenomena lead to new and interesting mathematical models. Here and everywhere in this book by a mathematical model we mean a system of partial differential equations, associated with boundary conditions and initial conditions, eventually, which describes a specific contact process.

The purpose of this book is to introduce the reader to some representative mathematical models which arise in Contact Mechanics. Our aim is twofold: first, to present a sound and rigorous description of the way in which the mathematical models are constructed; second, to present the mathematical analysis of such models which includes the variational formulation, existence, uniqueness and convergence results. To this end, we use results on various classes of variational inequalities in Hilbert spaces, that we present in an abstract functional framework. Also, we use various functional methods, including monotonicity, compactness, penalization, regularization and duality methods. Moreover, we pay particular attention to the mechanical interpretation of our results and, in this way, we illustrate the cross fertilization between modelling and applications on the one hand, and nonlinear analysis on the other hand.

This book is intended as a unified and readily accessible source for graduate students, as well as mathematicians, engineers and scientists. Its reading requires only basic knowledge of linear algebra, general topology, functional analysis and mechanics of continua.

In this chapter we illustrate the use of the abstract results obtained in Chapters 2 and 3 in the study of three frictionless or frictional contact problems with piezoelectric bodies. We model the material's behavior with an electro-elastic, an electro-viscoelastic and an electro-viscoplastic constitutive law, respectively. The contact is either bilateral or modelled with the normal compliance condition, with or without unilateral constraint. The friction is modelled with versions of Coulomb's law. The foundation is assumed to be either an insulator or electrically conductive. For each problem we provide a variational formulation which is in the form of a nonlinear system in which the unknowns are the displacement field and the electric potential field. Then we use the abstract existence and uniqueness results presented in Chapters 2 and 3 to prove the unique weak solvability of the corresponding contact problems. For the electro-elastic problem we also provide a dual variational formulation in terms of the stress and electric displacement fields. Everywhere in this chapter we consider the physical setting and the notation presented in Section 4.5, as well as the function spaces introduced in Section 4.1.

An Electro-elastic frictional contact problem

In this section we consider a frictional contact problem for electro-elastic materials. The problem is static and, therefore, we investigate it by using the arguments of elliptic variational inequalities presented in Section 2.2.

Problem statement

We assume that the body is electro-elastic and the foundation is an insulator.

In Chapter 2, we showed that flow disturbances can be decomposed into fluctuations in vorticity, entropy, and volume. The next two chapters focus on the evolution of vorticity in flows, with particular emphasis on the development of coherent vortical structures. Such large-scale structures, embedded on a background of acoustic waves and broadband, smaller-scale turbulence, dominate the unsteady flow fields in combustors. These large-scale structures play important roles in processes such as combustion instabilities, mixing and entrainment, flashback, and blowoff. For example, we will discuss vortex–flame interactions repeatedly in discussions of combustion instabilitiesin later chapters.

High Reynolds number flows are effectively inviscid outside the boundary layer. Vorticity in the flow largely originates, then, from boundary layers in approach flow passages or other walls. Free shear layers arise at points of boundary layer separation, initiating a sequence of large-scale flow instabilities as this vorticity is then stretched and amplified by the base flow.

Coherent structures arise because the base flow configuration, u0(y) or ū(y), is unstable. Many of the same instabilities that are focused on in the next two chapters that play such important roles in unsteady combustor dynamics also manifest themselves in a variety of other instances, including in spectacular, large-scale fashion in nature. To illustrate, Figure 3–1(a) illustrates cloud patterns showing the Kelvin- Helmholtz instability, discussed in Sections 3.4 and 4.1. Figure 3–1(b) illustrates the Bénard/von Kármán instability over a Japanese island visualized from space, to be discussed in Section 4.2.

This chapter initiates the third section of the text, discussing transient and time-harmonic combustor phenomena. This particular chapter focuses on the transient phenomena of flashback, flame stabilization, and blowoff. Chapters 11 and 12 then focus on time-harmonic and broadband flame forcing.

This chapter is divided into two main sections. Section 10.1 treats flame flashback. It shows that there are multiple mechanisms through which a flame can propagate upstream into premixed reactants, each of which has different sensitivities to the flow field and operating conditions. We also show that the processes controlling the initiation of flashback, and those controlling its behavior once it has begun to propagate upstream, are quite different. Section 10.2 then treats flame stabilization and blowoff. This chapter starts with the classical treatment of the problem, by considering the relative balance between flame speed and flow velocity in the shear layer. However, flames are strongly affected by stretch effects near the stabilization point, as they lie in regions of high shear. As such, we then work out the scalings for flame stretch rate in a shear layer and show that quite different results are possible, depending on the configuration. We also discuss effects of flow recirculation on flame stabilization and the processes leading to blowoff.

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.