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.

From a historical viewpoint, the centrifugal compressor configuration was developed and used well before axial-flow compressors, even in the propulsion field. The common belief that such a “bulky” compressor type, because of its large envelope and weight (Fig. 11.1), has no place except in ground applications is not exactly accurate. For example, with a typical total-to-total pressure ratio of, say, 5:1, it would take up to three axial-compressor stages to absorb similar amounts of shaft work that a single centrifugal compressor stage would. In fact, the added engine length, with so many axial stages, would increase the skin-friction drag on the engine exterior almost as much as the profile drag, which is a function of the frontal area.

Despite the preceding argument, the tradition remains that the centrifugalcompressor propulsion applications are unpopular. Exceptions to this rule include turboprop engines and short-mission turbofan engines, as shown in Figure 11.2.

An attractive feature of centrifugal compressors has to do with their off-design performance. Carefully designed, a centrifugal compressor will operate efficiently over a comparatively wider shaft speed range. This exclusive advantage helps alleviate some of the problems associated with the turbine-compressor matching within the gas generator.

One of the inherent drawbacks of centrifugal compressors has to do with multiple staging. As illustrated in Figure 2.12, the excessive 180 flow-turning angle of the annular return duct, in this case, will increase the flow rotationality (in terms of vorticity) and encourage the cross-stream secondary flow migration. This simply sets the stage for high magnitudes of total pressure loss and boundary-layer separation.

Historically, the first axial turbine utilizing a compressible fluid was a steam turbine. Gas turbines were later developed for engineering applications where compactness is as important as performance. However, the successful use of this turbine type had to wait for advances in the area of compressor performance. The viability of gas turbines was demonstrated upon developing special alloys that possess high strength capabilities at exceedingly high turbine-inlet temperatures.

In the history of axial turbines, most of the experience relating to the behavior of steam-turbine blading was put to use in gas-turbine blading and vice versa. This holds true as long as the steam remains in the superheated phase and not in the wet-mixture zone, for the latter constitutes a two-phase flow with its own problems (e.g., liquid impingement forces and corrosion).

Stage Definition

Figure 8.1 shows an axial-flow turbine stage consisting of a stator that is followed by a rotor. Figure 8.2 shows a hypothetical cylinder that cuts through the rotor blades at a radius that is midway between the hub and tip radii. The unwrapped version of the cylindrical surface in this figure is that where the stage inlet and exit velocity triangles will be required. As has been the terminology in preceding chapters, a stator airfoil will be termed a vane, and the rotor cascade consists of blades. An axial-flow turbine operating under a high (inlet-over-exit) pressure ratio would normally consist of several stages, each of the type shown in Figure 8.1, in an arrangement where the annulus height is rising in the through-flow direction (Fig. 8.3).