A brief introduction to gas-turbine engines was presented in Chapter 1. A review of the different engines, included in this chapter, reveals that most of these engine components are composed of “lifting” bodies, termed airfoil cascades, some of which are rotating and others stationary. These are all, by necessity, bound by the hub surface and the engine casing (or housing), as shown in Figures. 2.1 through 2.5. As a result, the problem becomes one of the internal-aerodynamics type, as opposed to such traditional external-aerodynamics topics as “wing theory” and others. Referring, in particular, to the turbofan engines in Chapter 1 (e.g., Fig. 1.3), these components may come in the form of ducted fans. These, as well as compressors and turbines, can be categorically lumped under the term “turbomachines.” Being unbound, however, the propeller of a turboprop engine (Fig. 1.2) does not belong to the turbomachinery category.

The turbomachines just mentioned, however, are no more than a subfamily of a more inclusive category. These only constitute the turbomachines that commonly utilize a compressible working medium, which is totally, or predominantly, air. In fact, a complete list of this compressible-flow subfamily should also include such devices as steam turbines, which may utilize either a dry (superheated) or wet (liquid/vapor) steam mixture with high quality (or dryness factor). However, there exists a separate incompressible-flow turbomachinery classification, where the working medium may be water or, for instance, liquid forms of oxygen or hydrogen, as is the case in the Space Shuttle Main Engine turbopumps. This subcategory also includes powerproducing turbomachines, such as water turbines.

Definition

A gas turbine engine is a device that is designed to convert the thermal energy of a fuel into some form of useful power, such as mechanical (or shaft) power or a highspeed thrust of a jet. The engine consists, basically, of a gas generator and a power-conversion section, as shown in Figures 1.1 and 1.2.

As is clear in these figures, the gas generator consists of the compressor, combustor, and turbine sections. In this assembly, the turbine extracts shaft power to at least drive the compressor, which is the case of a turbojet. Typically, in most other applications, the turbine will extract more shaft work by comparison. The excess amount in this case will be transmitted to a ducted fan (turbofan engine) or a propeller (turboprop engine), as seen in Figure 1.2. However, the shaft work may also be utilized in supplying direct shaft work, or producing electricity in the case of a power plant or an auxiliary power unit (Fig. 1.1). The fact, in light of Figures 1.1 through 1.5, is that different types of gas-turbine engines clearly result from adding various inlet and exit components, to the gas generator. An always interesting component in this context is the thrust-augmentation devices known as afterburners in a special class of advanced propulsion systems (Fig. 1.4).

Gas-turbine engines are exclusively used to power airplanes because of their high power-to-weight ratio. They have also been used for electric-power generation in pipeline-compressor drives, as well as to propel trucks and tanks. In fact, it would be unwise to say that all possible turbomachinery applications have already been explored.

The most powerful insights into the behavior of the physical world are obtained when observations are well described by a theoretical framework that is then available for predicting new phenomena or new observations. An example is the observed behavior of radio signals and their extremely accurate description by the Maxwell equations of electromagnetic radiation. Other such examples include planetary motions through Newtonian mechanics, or the movement of the atmosphere and ocean as described by the equations of fluid mechanics, or the propagation of seismic waves as described by the elastic wave equations. To the degree that the theoretical framework supports, and is supported by, the observations one develops sufficient confidence to calculate similar phenomena in previously unexplored domains or to make predictions of future behavior (e.g., the position of the moon in 1000 years, or the climate state of the earth in 100 years).

Developing a coherent view of the physical world requires some mastery, therefore, of both a framework, and of the meaning and interpretation of real data. Conventional scientific education, at least in the physical sciences, puts a heavy emphasis on learning how to solve appropriate differential and partial differential equations (Maxwell, Schrödinger, Navier—Stokes, etc.). One learns which problems are “well-posed,” how to construct solutions either exactly or approximately, and how to interpret the results.

The focus will now shift away from discussion of estimation methods in a somewhat abstract context, to more specific applications, primarily for large-scale fluid flows, and to the ocean in particular. When the first edition of this book (OCIP) was written, oceanographic uses of the methods described here were still extremely unfamiliar to many, and they retained an aura of controversy. Controversy arose for two reasons: determining the ocean circulation was a classical problem that had been discussed with ideas and methods that had hardly changed in 100 years; the introduction of algebraic and computer methods seemed to many to be an unwelcome alien graft onto an old and familiar problem. Second, some of the results of the use of these methods were so at odds with “what everyone knew,” that those results were rejected out of hand as being obviously wrong – with the methods being assumed flawed.

In the intervening 25+ years, both the methodology and the inferences drawn have become more familiar and less threatening. This change in outlook permits the present chapter to focus much less on the why and how of such methods in the oceanographic context, and much more on specific examples of how they have been used. Time-dependent problems and methods will be discussed in Chapter 7.

This book is to a large extent the second edition of The Ocean Circulation Inverse Problem, but it differs from the original version in a number of ways. While teaching the basic material at MIT and elsewhere over the past ten years, it became clear that it was of interest to many students outside of physical oceanography — the audience for whom the book had been written. The oceanographic material, instead of being a motivating factor, was in practice an obstacle to understanding for students with no oceanic background. In the revision, therefore, I have tried to make the examples more generic and understandable, I hope, to anyone with even rudimentary experience with simple fluid flows.

Also many of the oceanographic applications of the methods, which were still novel and controversial at the time of writing, have become familiar and almost commonplace. The oceanography, now confined to the two last chapters, is thus focussed less on explaining why and how the calculations were done, and more on summarizing what has been accomplished. Furthermore, the time-dependent problem (here called “state estimation” to distinguish it from meteorological practice) has evolved rapidly in the oceanographic community from a hypothetical methodology to one that is clearly practical and in ever-growing use.

Background

The purpose of this chapter is to record a number of results that are useful in finding and understanding the solutions to sets of usually noisy simultaneous linear equations and in which formally there may be too much or too little information. A lot of the material is elementary; good textbooks exist, to which the reader will be referred. Some of what follows is discussed primarily so as to produce a consistent notation for later use. But some topics are given what may be an unfamiliar interpretation, and I urge everyone to at least skim the chapter.

Our basic tools are those of matrix and vector algebra as they relate to the solution of linear simultaneous equations, and some elementary statistical ideas — mainly concerning covariance, correlation, and dispersion. Least-squares is reviewed, with an emphasis placed upon the arbitrariness of the distinction between knowns, unknowns, and noise. The singular-value decomposition is a central building block, producing the clearest understanding of least-squares and related formulations. Minimum variance estimation is introduced through the Gauss—Markov theorem as an alternative method for obtaining solutions to simultaneous equations, and its relation to and distinction from least-squares is discussed. The chapter ends with a brief discussion of recursive least-squares and estimation; this part is essential background for the study of time-dependent problems in Chapter 4.