Introduction

When two engineering surfaces are loaded together there will always be some distortion of each of them. These deformations may be purely elastic or may involve some additional plastic, and so permanent, changes in shape. Such deflections and modifications in the surface profiles of the components can be viewed at two different scales. For example, consider the contact between a heavily loaded roller and the inner and outer races in a rolling-element bearing. In examining the degree of flattening of the roller we could choose to express the deflections as a proportion of their radii, that is, to view the distortions on a relatively macroscopic scale. On the other hand, as we have seen in Chapter 2, at the microscale no real surface, such as those of the roller or the race, can be truly smooth, and so it follows that when these two solid bodies are pushed into contact they will touch initially at a discrete number of points or asperities. The sum of the areas of all these contact spots, the ‘true’ area of contact, will be a relatively small proportion of the ‘nominal’ or geometric contact area–perhaps as little as only a few per cent of it. Some deformation of the material occurs on a very small scale at, or very close to, these areas of true contact. It is within these regions that the stresses are generated whose total effect is just to balance the applied load.

Introduction

A hydrostatic bearing is one in which the loaded surfaces are separated by a fluid film which is forced between them by an externally generated pressure. Formation of the film, and so successful operation of the bearing, requires the supply pump to operate continuously, but it does not depend on the relative motion of the surfaces (hence the term ‘hydrostatic’). Such bearings have a great attraction to engineers; machine elements supported in this way move with incomparable smoothness and the only restriction to motion arises from the small viscous losses in the fluid. A mass supported on a hydrostatic bearing will glide silently down the slightest incline.

The essential features of a typical hydrostatic single-pad thrust bearing are shown in Fig. 6.1 (a). The bearing is supplied with fluid under pressure ps which, before entering the central pocket or recess, passes through some form of restrictor or compensator in which its pressure is dropped to some lower value pr. The fluid then passes out of the bearing through the narrow gap, shown of thickness h, between the bearing land and the opposing bearing surface or slider which is also often known as the bearing runner. The depth of the pocket is very much greater than the gap h. The restrictor is an essential feature of the bearing since it allows the pocket pressure pr to be different from the supply pressure; this difference, between pr and ps, depends on the load applied W.

Introduction

In many internal flows there are only limited regions in which the velocity can be considered irrotational; i.e. in which the motion is such that particles travel without local rotation. In an irrotational, or potential, flow the velocity can be expressed as the gradient of a scalar function. This condition allows great simplification and, where it can be employed, is of enormous utility. Although we have given examples of its use, potential flow theory has a narrower scope in internal flow than in external flow and the description and analysis of non-potential, or rotational, motions plays a larger role in the former than in the latter. One reason for this difference is the greater presence of bounding solid surfaces and the accompanying greater opportunity for viscous shear forces to act. Even in those internal flow configurations in which the flow can be considered inviscid, however, different streamtubes can receive different amounts of energy (from fluid machinery, for example), resulting in velocity distributions which do not generally correspond to potential flows. Because of this, we now examine two key fluid dynamic concepts associated with rotational flows: vorticity, which has to do with the local rate of rotation of a fluid particle, and circulation, a related, but more global, quantity.

Before formally introducing these concepts, it is appropriate to give some discussion concerning the motivation for working with them, rather than velocity and pressure fields only. The equations of motion for a fluid contain expressions of forces and acceleration, derived from Newton's laws.

Introduction

In this chapter we address three-dimensional flows in which streamwise vorticity is a prominent feature. Three main topics are discussed. The first, and principal, subject falls under the general label of secondary flows, cross-flow plane (secondary) circulations which occur in flows that were parallel at some upstream station. The second is the enhancement of mixing by embedded streamwise vorticity and the accompanying motions normal to the bulk flow direction (see for example Bushnell (1992)). The third is the connection between vorticity generation and fluid impulse.

The different topics are linked in at least three ways. First, the class of fluid motions described are truly three-dimensional. Second, focus on the vortex structure in these flows is a way to increase physical insight. The perspective of the chapter is that the flows of interest are rotational and three-dimensional, and the appropriate tools for capturing their quantitative behavior are three-dimensional numerical simulations (e.g. Launder (1995)). Results from such computations, as well as from experiments, are used to illustrate the overall features. To complement detailed simulations and experiments, however, it is often helpful to have a simplified description of the motion which can guide the interrogation and scope of the computations, enable understanding of why different effects are seen, and suggest scaling for different mechanisms. The ideas about vorticity evolution and vortex structure, introduced in Chapter 3, provide a skeleton for this type of description.

Introduction

In this chapter the discussion of fluid component and system response to disturbances, begun in Chapter 6, is extended to a broader class of flow non-uniformities. Whereas Chapter 6 considered primarily one-dimensional disturbances, that restriction is now dropped and we address more general (two- and three-dimensional) non-uniformities with variations transverse to the bulk flow direction. Examples of interest are turbomachines subjected to circumferentially varying inlet conditions and the behavior of components with geometry generated non-uniformity, such as is caused by a contraction or a bend in close proximity.

Three important issues relating to these situations can be identified. One is the effect of the fluid component on the flow non-uniformity, or distortion: how are the non-uniformities altered by passage through the component? A second is the effect of the non-uniformity on the component: how does the distortion modify the component performance? The approaches needed to address these two questions are fundamentally different. For the former, qualitative aspects, and even many quantitative features, can be resolved within the framework of a linearized description. For the latter, however, the problem is inherently nonlinear and a different level of analysis is needed. Beyond component performance there is a third issue. Because fluid components typically occur as part of an overall system, what changes in interactions with the rest of the system arise due to the non-uniformity?

Several integrating themes thread through the different applications discussed. The first is that fluid components do not passively accept non-uniform flow but play a major role in modifying the velocity distribution.

There are a number of excellent texts on fluid mechanics which focus on external flow, flows typified by those around aircraft, ships, and automobiles. For many fluid devices of engineering importance, however, the motion is appropriately characterized as an internal flow. Examples include jet engines or other propulsion systems, fluid machinery such as compressors, turbines, and pumps, and duct flows, including nozzles, diffusers, and combustors. These provide the focus for the present book.

Internal flow exhibits a rich array of fluid dynamic behavior not encountered in external flow. Further, much of the information about internal flow is dispersed in the technical literature and does not appear in a connected treatment that is accessible to students as well as to professional engineers. Our aim in writing this book is to provide such a treatment.

A theme of the book is that one can learn a great deal about the behavior of fluid components and systems through rigorous use of basic principles (the concepts). A direct way to make this point is to present illustrations of technologically important flows in which it is true (the applications). This link between the two is shown in a range of internal flow examples, many of which appear for the first time in a textbook.

The experience of the authors spans dealing with internal flow in an industrial environment, teaching the topic to engineers in industry and government, and teaching it to students at MIT. The perspective and selection of material reflects (and addresses) this span.

Introduction

This is a book about the fluid motions which set the performance of devices such as propulsion systems and their components, fluid machinery, ducts, and channels. The flows addressed can be broadly characterized as follows:

1. There is often work or heat transfer. Further, this energy addition can vary between streamlines, with the result that there is no “uniform free stream”. Stagnation conditions therefore have a spatial (and sometimes a temporal) variation which must be captured in descriptions of the component behavior.

2. There are often large changes in direction and in velocity. For example, deflections of over 90° are common in fluid machinery, with no one obvious reference direction or velocity. Concepts of lift and drag, which are central to external aerodynamics, are thus much less useful than ideas of loss and flow deflection in describing internal flow component performance. Deflection of the non-uniform flows mentioned in (1) also creates (three-dimensional) motions normal to the mean flow direction which transport mass, momentum, and energy across ducts and channels.

3. There is often strong swirl, with consequent phenomena that are different than for flow without swirl. For example, static pressure rise can be associated almost entirely with the circumferential (swirl) velocity component and thus essentially independent of whether the flow is forward (radially outward) or separated (radially inward). In addition the upstream influence of a fluid component, and hence the interaction between fluid components in a given system, can be qualitatively different than that in a flow with no swirl.

4. […]

Introduction

Efficiency can be the most important parameter for many fluid machines and characterizing the losses which determine the efficiency is a critical aspect in the analysis of these devices. This chapter describes basic mechanisms for loss creation in fluid flows, defines the different measures developed for assessing loss, and examines their applicability in various situations.

In external aerodynamics, drag on an aircraft or vehicle is most frequently the measure of performance loss. The product of drag and forward velocity represents the power that has to be supplied to drive the vehicle. Defining drag, however, requires defining the direction in which it acts and determining the power expended requires specification of an appropriate velocity. The choice of direction is clear for most external flows but it is less evident in internal flows. Within gas turbine engines, for example, there are situations in which viscous forces can be nearly perpendicular to the mean stream direction or in which the mean stream direction changes by as much as 180°, as in a reverse flow combustor. There is also some ambiguity in the choice of an appropriate reference velocity for power input, even in simple internal flow configurations such as nozzles or diffusers where the velocity changes from inlet to outlet.

Because of this, the most useful indicator of performance loss and inefficiency in internal flows is the entropy generated due to irreversibility. The arguments that underpin this statement are presented in the first part of the chapter to illustrate quantitatively the connection between entropy rise andwork lost through an irreversible process.

Introduction

In the analysis of fluid machinery behavior, it is often advantageous to view the flow from a coordinate system fixed to the rotating parts. Adopting such a coordinate system allows one to work with fluid motions which are steady, but there is a price to be paid because the rotating system is not inertial. In an inertial coordinate system, Newton's laws are applicable and the acceleration on a particle of mass m is directly related to the vector sum of forces through F = ma. In a rotating coordinate system, the perceived accelerations also include the Coriolis and centrifugal accelerations which must be accounted for if we wish to write Newton's second law with reference to the rotating system.

In this chapter we examine flows in rotating passages (ducts, pipes, diffusers, and nozzles). These typically operate in a regime where rotation has an effect on device performance but does not dominate the behavior to the extent found in the geophysical applications which are considered in much of the literature (e.g. Greenspan (1968)). The objectives are to develop criteria for when phenomena associated with rotation are likely to be important and to illustrate the influence of rotation on overall flow patterns. A derivation of the equations of motion in a rotating frame of reference is first presented to show the origin of the Coriolis and centrifugal accelerations, with illustrations provided of the differences between flow as seen in fixed (often called absolute) and rotating (often called relative) systems. Quantities that are conserved in a steady rotating flow are then discussed, because these find frequent use in fluid machinery.

Introduction

This chapter introduces a variety of basic ideas encountered in analysis of internal flow problems. These concepts are not only useful in their own right but they also underpin material which appears later in the book.

The chapter starts with a discussion of conditions under which a given flow can be regarded as incompressible. If these conditions are met, the thermodynamics have no effect on the dynamics and significant simplifications occur in the description of the motion.

The nature and magnitude of upstream influence, i.e. the upstream effect of a downstream component in a fluid system, is next examined. A simple analysis is developed to determine the spatial extent of such influence and hence the conditions under which components in an internal flow system are strongly coupled.

Many flows of interest cannot be regarded as incompressible so that effects associated with compressibility must be addressed. We therefore introduce several compressible flow phenomena including one-dimensional channel flow, mass flow restriction (“choking”) at a geometric throat, and shock waves. The last of these topics is developed first from a control volume perspective and then through a more detailed analysis of the internal shock structure to show how entropy creation occurs within the control volume.

The integral forms of the equations of motion, utilized in a control volume formulation, provide a powerful tool for obtaining an overall description of many internal flow configurations. A number of situations are analyzed to show their application. These examples also serve as modules for building descriptions of more complex devices.

Introduction

Many fluid machinery applications involve swirling flow. Devices in which swirl phenomena have a strong influence include combustion chambers, turbomachines and their associated ducting, and cyclone separators. In this chapter, we examine five aspects of swirling flows: (i) an introductory description of pressure and velocity fields in these types of motion; (ii) the increased capability for downstream conditions to affect upstream flow; (iii) instabilities and propagating waves on vortex cores; (iv) the behavior of vortex cores in pressure gradients; and (v) viscous swirling flow, specifically the influence of swirl on boundary layers, jets, mixing, and recirculation. The behavior of vortex cores ((iii) and (iv)) is described in some depth because this type of embedded structure features in a number of fluid devices. Further, much of the focus is on inviscid flow because the dominant effects of swirl are inertial in nature.

In the discussion it is necessary to modify some of the concepts developed for non-swirling flow. For example, there can be a large variation in static pressure through a vortex core at the center of a swirling flow, in contrast to the essentially uniform static pressure across a thin shear layer or boundary layer in a flow with no swirl. This pressure variation affects the vortex core evolution. The length scales which characterize the upstream influence of a fluid component are also altered when swirl exists.

Different parameters exist in the literature for representing the swirl level in a given flow. These have been developed to enable the definition of flow regimes and behavior.