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.