Fluid dynamics and solid mechanics

Summary

This chapter discusses the application of Pesin theory to linked twist maps. Drawing on three key papers from the ergodic theory literature we give the proof that linked twist maps can be decomposed into at most a countable number of ergodic components.

Introduction

In Chapter 4 we gave Devaney's construction of a horseshoe for a linked twist map on the plane. The existence of the horseshoe and the accompanying subshift of finite type implies that the linked twist map contains a certain amount of complexity. However, topological features such as horseshoes may not be of interest from a statistical, observable, or measure-theoretic point of view, as they occur on invariant sets of measure zero. The subshift of finite type occurs on just such an invariant set of measure zero and is therefore arguably not of significant statistical interest. Nevertheless it is possible that similar behaviour is shared by points in the vicinity of the horseshoe, meaning that complex behaviour is present in a significant (that is, positive measure) domain. Easton (1978) conjectures that this may indeed be the case, and that in fact linked twist maps may be ergodic.

Three papers provide the framework for applying the results of Pesin (1977) connecting hyperbolicity and ergodicity. In this and the following two chapters we draw heavily on each of Burton & Easton (1980), Wojtkowski (1980) and Przytycki (1983).

Summary

In this chapter we give formal definitions of linked twist maps on the plane and linked twist maps on the torus. We give heuristic descriptions of the mechanisms that give rise to good mixing for linked twist maps, and highlight the role played by ‘co-rotation’ and ‘counter-rotation’. We show how to construct linked twist maps from blinking flows and from duct flows, and we describe a number of additional examples of mixers that can be treated within the linked twist map framework.

Introduction

The central theme of this book is that the mathematical notion of a linked twist map, and attendant dynamical consequences, is naturally present in a variety of different mixing situations. In this chapter we will define what we mean by a linked twist map, and then give a general idea of why they capture the essence of ‘good mixing’. To do this we will first describe the notion of a linked twist map as first studied in the mathematical literature. This setting may at first appear to have little to do with the types of situations arising in fluid mechanics, but we will argue the contrary later. However, this more mathematically ideal setting allows one to rigorously prove strong mixing properties in a rather direct fashion that would likely be impossible for the types of maps arising in typical fluid mechanical situations. We will then consider a variety of mixers and mixing situations and show how the linked twist map structure naturally arises.

Summary

This chapter contains concepts and results from the field of hyperbolic dynamical systems. We define uniform and nonuniform hyperbolicity, and go on to describe Pesin theory, which creates a bridge between nonuniform hyperbolicity and the ergodic hierarchy.

Introduction

Hyperbolic dynamics, loosely speaking, concerns the study of systems which exhibit both expanding and contracting behaviour. Hyperbolicity is one of the most fundamental aspects of dynamical systems theory, both from the point of view of pure dynamical systems, in which it represents a widely studied and thoroughly understood class of system, and from the point of view of applied dynamical systems, in which it gives one of the simplest models of complex and chaotic dynamics. However the pay-off for this amount of knowledge and (apparent) simplicity is severe. While hyperbolic objects (for example certain fixed points and periodic orbits, and horseshoes, like that constructed in the previous chapter) are common enough occurrences, these are arguably of limited practical importance, as all these objects comprise sets of zero (Lebesgue) measure. There are only a handful of real systems for which the strongest form of hyperbolicity (uniform hyperbolicity) has been shownto exist on a set of positive measure. Typically, uniformly hyperbolic systems tend to be restricted to model systems, such as the Arnold Cat Map (Arnold & Avez (1968)), or idealized mechanical examples, such as the triple linkage of Hunt & Mackay (2003).

Weaker forms of hyperbolicity have been studied in great detail, and powerful results exist linking these to mixing properties, but still any sort of hyperbolicity is not a straightforward property to demonstrate.

Summary

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.

Summary

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).

Summary

In Chapters 3 and 4, we studied major changes in the thermophysical properties of a flow as it traverses a turbine or compressor stage. The analysis then was onedimensional, with the underlying assumption that average flow properties will prevail midway between the endwalls. Categorized as a pitch-line flow model, this “bulk-flow” analysis proceeds along the “master” streamline (or pitch line), with no attention given to any lateral flow-property gradients.

However, we know of, at least, one radius-dependent variable, namely the tangential “solid-body” velocity vector (U). The question addressed in this chapter is how the other thermophysical properties vary along the local annulus height at any streamwise location. The stator and rotor inlet and exit stations, being important “control” locations, are particularly important in this context. In the following, the so-called radial equilibrium equation is derived and specific simple solutions offered. Despite the flow-model simplicity, the radial-equilibrium equation enables the designer early on to take a look at preliminary magnitudes of such important variables as the hub and tip reactions prior to the detailed design phase.

Assumptions

For any axial stator-to-rotor or interstage gap in Figure 6.1, the following assumptions are made:

Summary

Beginning with the class-notes version, this book is the outcome of teaching the courses of Principles of Turbomachinery and Aerospace Propulsion in the mechanical and aerospace engineering departments of Texas A&M University. Over a period of fourteen years, the contents were continually altered and upgraded in light of the students' feedback. This has always been insightful, enlightening, and highly constructive.

The book is intended for junior- and senior-level students in the mechanical and aerospace engineering disciplines, who are taking gas-turbine or propulsion courses. In its details, the text serves the students in two basic ways. First, it refamiliarizes them with specific fundamentals in the fluid mechanics and thermodynamics areas, which are directly relevant to the turbomachinery design and analysis aspects. In doing so, it purposely deviates from such inapplicable subtopics as external (unbound) flows around geometrically standard objects and airframe-wing analogies. Instead, turbomachinery subcomponents are utilized in such a way to impart the element of practicality and highlight the internal-flow nature of the subject at hand. The second book task is to prepare the student for practical design topics by placing him or her in appropriate real-life design settings. In proceeding from the first to the second task, I have made every effort to simplify the essential turbomachinery concepts, without compromising their analytical or design-related values.

Judging by my experience, two additional groups are served by the book. First, practicing engineers including, but not necessarily limited to, those at the entry level. As an example, the reader in this category will benefit from the practical means of estimating the stage aerodynamic losses.

Summary

Introduction

The utilization of axial-flow compressors (Fig. 9.1) in gas-turbine engines has been relatively recent. The history of this compressor type began following an era when centrifugal compressors (Fig. 9.2) were dominant. It was later confirmed, on an experimental basis, that axial-flow compressors can run much more efficiently. Earlier attempts to build multistage axial-flow compressors entailed running multistage axial-flow turbines in the reverse direction. As presented in Chapter 4, a compressor-stage reaction in this case will be negative, a situation that has its own performance-degradation effect. Today, carefully designed axial-flow compressor stages can very well have efficiencies in excess of 80%. A good part of this advancement is because of the standardization of thoughtfully devised compressor-cascade blading rules.

Comparison with Axial-Flow Turbines

In passing through a reaction-turbine blade row, the flow stream will continually lose static pressure and enthalpy. The result is a corresponding rise in kinetic energy, making the process one of the flow-acceleration type. In axial compressors, by contrast, an unfavorable static pressure gradient prevails under which large-scale losses become more than likely. It is therefore sensible to take greater care in the compressor-blading phase. Another major difference, by reference to Figure 9.3, is that the compressor meridional flow path is geometrically converging as opposed to the typically diverging flow path of a turbine. This is a direct result of the streamwise density rise in this case, as will be discussed later. Referring to Figure 9.4, another difference in this context is a substantially greater blade count compared wth axial-turbine rotors, where the number of blades is typically in the low twenties.

Summary

In this chapter, the flow-governing equations (so-called conservation laws) are reviewed, with applications that are purposely turbomachinery-related. Particular emphasis is placed on the total (or stagnation) flow properties. A turbomachinery-adapted Mach number definition is also introduced as a compressibility measure of the flow field. A considerable part of the chapter is devoted to the so-called total relative properties, which, together with the relative velocity, define a legitimate thermophysical state. Different means of gauging the performance of a turbomachine, and the wisdom behind each of them, are discussed. Also explored is the entropy-production principle as a way of assessing the performance of turbomachinery components. The point is stressed that entropy production may indeed be desirable, for it is the only meaningful performance measure that is accumulative (or addable) by its mere definition.

The flow behavior and loss mechanisms in two unbladed components of gas turbines are also presented. The first is the stator/rotor and interstage gaps in multistage axial-flow turbomachines. The second component is necessarily part of a turboshaft engine. This is the exhaust diffuser downstream from the turbine section. The objective of this component is to convert some of the turbine-exit kinetic energy into a static pressure rise. Note that it is by no means unusual for the turbine-exit static pressure to be less than the ambient magnitude, which is where the exhaust-diffuser role presents itself.

In terms of the flow-governing equations, two nonvectorial equations will be covered in this chapter. These are the energy- and mass-conservation equations (better known as the First Law of Thermodynamics and the continuity equation, respectively).

Summary

Introduction

In this chapter, a turbomachinery-related nondimensional groupings of geometrical dimensions and thermodynamic properties will be derived. These will aid us in many tasks, such as:

Fluid dynamics and solid mechanics

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8126 results in Fluid dynamics and solid mechanics

6 - The ergodic partition for toral linked twist maps

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2 - Linked twist maps: definition, construction and the relevance to mixing

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5 - Hyperbolicity

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Index

Acknowledgments

References

Frontmatter

11 - Centrifugal Compressors

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8 - Axial-Flow Turbines

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Frontmatter

7 - Polytropic (Small-Stage) Efficiency

6 - Radial-Equilibrium Theory

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4 - Energy Transfer between a Fluid and a Rotor

Contents

Preface

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10 - Radial-Inflow Turbines

9 - Axial-Flow Compressors

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12 - Turbine-Compressor Matching

3 - Aerothermodynamics of Turbomachines and Design-Related Topics

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5 - Dimensional Analysis, Maps, and Specific Speed

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