In Chapter 1, we discussed the equilibrium states of a thermochemical system. We have shown that by knowing the initial state of a reactive mixture, we can determine its final state after chemical and thermal equilibria have been established. However, the equilibrium calculations are not able to answer such relevant questions as: how does the mixture get from the initial state to the final state, and how long does it take to do so? Obviously if a particular reaction proceeds exceedingly slowly compared to other physical or chemical processes of interest, it is likely that this reaction could be either irrelevant to the system's behavior or not of controlling importance in being the rate-limiting step in the system's evolution.

In this chapter, we first present the phenomenological law describing the general dependence of reaction rates on reactant concentrations and temperature. We then discuss multistep reactions and some approximation techniques used to simplify the representation of these reactions.

In Section 2.2, we present the specific functional dependence of the reaction rate on temperature—the Arrhenius law. We then derive and discuss three theories of reaction rates. The collision theory is based on the kinetic theory of gases, counting the frequency of molecular collisions that are energetic enough to cause the colliding molecules to react. The transition state theory examines the activated state of molecules and derives the reaction rate by considering the characteristic times and energies associated with their transition from activated to reacted states.

In almost all the combustion problems studied so far, the flows are sufficiently subsonic such that M ≪ 1, where M is the Mach number. There are, however, situations in which reactions take place in flows whose velocities can be sufficiently high such that they either are close to sonic or are supersonic. Examples are combustion within supersonic ramjet (scramjet) engines and the initiation and propagation of blast waves.

In nonreactive fluid mechanics such high-speed flows are called compressible flows because density now varies appreciably with the flow velocity. Such a density variation is to be distinguished from that caused by the large amount of heat release in reactive, low subsonic flows studied in previous chapters. It is therefore important to recognize that density can still vary significantly in an aerodynamically incompressible, low subsonic flow due to heat release. When it is actually assumed to be constant for such a flow, either due to the smallness of the heat release or for analytical expediency, then the flow is said to be one of constant density.

There are several fundamental differences between high-speed flows and low subsonic flows. First, the isobaric assumption of Section 5.2.4 ceases to hold. Second, the kinetic energy of the flow is now appreciable as compared to the chemical energy and frequently needs to be considered. Indeed, we have already encountered this issue in Section 12.5 on supersonic boundary-layer flows. Third, while diffusion is an essential process in low subsonic flows, convection frequently dominates over diffusion in high-speed flows except for situations involving steep gradients such as those within boundary layers.

We now begin the study of premixed combustion. As we have learned from Chapter 6, a nonpremixed flame is supported by the stoichiometric, counterdiffusion of fuel and oxidizer. Thus, once ignited, a nonpremixed flame will situate itself somewhere between the fuel and oxidizer sources in order to satisfy this stoichiometry requirement. However, once ignition is achieved in a combustible fuel–oxidizer mixture, the resulting premixed flame tends to propagate into and consume the unburned mixture, if unrestrained through some aerodynamic means. Thus a premixed flame is a wave phenomenon.

In this chapter we shall study the simplest, idealized mode of wave propagation, namely the steady propagation of a one-dimensional, planar, adiabatic, wave relative to a stationary, combustible mixture in the doubly infinite domain. We shall call such a wave a standard wave or standard flame. In Section 7.1 we shall identify all such possible waves by constraining, through the conservation of mass, momentum, and energy, the states far upstream and downstream of the wave where the nonequilibrium processes of diffusion and reaction both vanish. Such an analysis yields the Rankine–Hugoniot relations, which show that two classes of waves can propagate in a combustible mixture, namely subsonic deflagration waves and supersonic, detonation waves. These waves have distinctively different properties.

Since the wave structure is not described at the level of the Rankine–Hugoniot analysis, the problem is not closed in that the crucial parameter of the wave response, namely the wave propagation speed, needs to be given.

GENERAL CONCEPTS

Most flows in practical combustion devices are turbulent, characterized by the presence of rapid, random fluctuations of the flow velocity and scalar properties at a given point in space. These fluctuations spread out in a manner similar to molecular diffusion as the flow evolves in time and/or proceeds downstream. Figure 11.1.1 illustrates various canonical flow configurations that are often encountered in practical combustion systems: unconfined flows such as jets and mixing layers, semiconfined flows over solid surfaces, confined flows in ducts, reverse flows in wakes, and buoyant flows.

Turbulence remains one of the most challenging and unsolved problems in physics. The complexity further increases when chemical reactions are also present. Because of these difficulties, studies on turbulent combustion have been mostly empirical until the late 1970s. Advances since then have identified fruitful paths for rational investigation. In this chapter we present a brief account of the current state of understanding.

In the next two sections the general concepts and solution techniques of turbulent flows, mostly nonreacting, are presented. These are followed by separate discussions on turbulent premixed and nonpremixed combustion. For a more detailed exposition, the reader is referred to Monin and Yaglom (1965), Tennekes and Lumley (1972), Launder and Spalding (1972), Hinze (1975), Schlichting et al. (1999), and Pope (2000) for nonreacting turbulent flows, and to Libby and Williams (1980, 1994), Williams (1985), Peters (2000), and Poinsot and Veynante (2005) for reacting turbulent flows.

Introduction

The remainder of this book focuses on numerical algorithms for the unsteady Euler equations in one dimension. Although the practical applications of the one-dimensional Euler equations are certainly limited per se, virtually all numerical algorithms for inviscid compressible flow in two and three dimensions owe their origin to techniques developed in the context of the one-dimensional Euler equations. It is therefore essential to understand the development and implementation of these algorithms in their original onedimensional context.

This chapter describes the principal mathematical properties of the onedimensional Euler equations. An understanding of these properties is essential to the development of numerical algorithms. The presentation herein is necessarily brief. For further details, the reader may consult, for example, Courant and Friedrichs (1948) and Landau and Lifshitz (1958).

Differential Forms of One-Dimensional Euler Equations

The one-dimensional Euler equations can be expressed in a variety of differential forms, of which three are particularly useful in the development of numerical algorithms. These forms are applicable where the flow variables are continuously differentiable. However, flow solutions may exhibit discontinuities that require separate treatment, as will be discussed later in Section 2.3.

The purpose of this book is to present the basic elements of numerical methods for compressible flows. The focus is on the unsteady one-dimensional Euler equations which form the basis for numerical algorithms in compressible fluid mechanics. The book is restricted to the basic concepts of finite volume methods, and even in this regards is not intended to be exhaustive in its treatment. Several noteworthy texts on numerical methods for compressible flows are cited herein.

I would like to express my appreciation to Florence Padgett and Peter Gordon (Cambridge University Press) and Robert Stengel (Princeton University) for their patience. Any omissions or errors are mine alone.