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.

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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.

Introduction

Chapters 10 and 11 address flows in which substantial changes in density occur. The changes arise from processes which are dynamical (e.g. density changes from pressure variations associated with fluid accelerations) or thermodynamic (density changes primarily from bulk heat addition due to chemical reaction or phase change) or a combination of the two. This chapter focuses primarily on situations with density variations due to dynamical effects; as we saw in Section 2.2, this means flows with Mach numbers significant compared to unity. Chapter 11 discusses flows with density variations primarily due to heat addition.

Much of the material is based on quasi-one-dimensional gas dynamics. Characterization of quasione- dimensional analysis as “the secret weapon of the internal fluid dynamicist” (Heiser, 1995) is an apt aphorism indeed. This type of treatment enables useful engineering estimates in a wide variety of situations and is a powerful tool for providing insight into the response of compressible flows to alterations in area, addition of mass, momentum, and energy, swirl, and flow non-uniformity. This is true not only for simple duct and channel flows but also for more complex problems, for example those arising in the matching of gas turbine engine components (Kerrebrock, 1992; Cumpsty, 1998).

Many computational techniques now exist to address internal flows in complex geometries. As such, we spend little time in discussion of approximations that were necessary in the past to attack compressible flow problems. One-dimensional analysis, however, is still very much a part of modern approaches to grappling with internal flow problems, even though its use as a detailed design tool has been supplanted by more accurate computations.

Introduction

Unsteady flow phenomena are important in fluid systems for several reasons. First is the capability for changes in the stagnation pressure and temperature of a fluid particle; the primary work interaction in a turbomachine is due to the presence of unsteady pressure fluctuations associated with the moving blades. A second reason for interest is associated with wave-like or oscillatory behavior, which enables a greatly increased influence of upstream interaction and component coupling through propagation of disturbances. The amplitude of these oscillations, which is set by the unsteady response of the fluid system to imposed disturbances, can be a limiting factor in defining operational regimes for many devices. A final reason is the potential for fluid instability, or self-excited oscillatory motion, either on a local (component) or global (fluid system) scale. Investigation of the conditions for which instability can occur is inherently an unsteady flow problem.

Unsteady flows have features quite different than those encountered in steady fluid motions. To address them Chapter 6 develops concepts and tools for unsteady flow problems.

The inherent unsteadiness of fluid machinery

To introduce the role unsteadiness plays in fluid machinery, consider flow through an adiabatic, frictionless turbomachine, as shown in Figure 6.1 (Dean, 1959). At the inlet and outlet of the device, and at the location where the work is transferred (by means of a shaft, say), conditions are such that the flow can be regarded as steady. We also restrict discussion to situations in which the average state of the fluid within the control volume is not changing with time.

Introduction

In this chapter, we discuss the types of thin shear layers that occur in flows in which the Reynolds number is large. The first of these is the boundary layer, or region near a solid boundary where viscous effects have reduced the velocity below the free-stream value. The reduced velocity in the boundary layer implies, as mentioned in Chapter 2, a decrease in the capacity of a channel or duct to carry flow and one effect of the boundary layer is that it acts as a blockage in the channel. Calculation of the magnitude of this blockage and the influence on the flow external to the boundary layer is one issue addressed in this chapter. Boundary layer flows are also associated with a dissipation of mechanical energy which manifests itself as a loss or inefficiency of the fluid process. Estimation of these losses is a focus of Chapter 5. The role of boundary layer blockage and loss in fluid machinery performance is critical; for a compressor or pump, for example, blockage is directly related to pressure rise capability and boundary layer losses are a determinant of peak efficiency that can be obtained.

Another type of shear layer is the free shear layer or mixing layer, which forms the transition region between two streams of differing velocity. Examples are jet or nozzle exhausts, mixing ducts in a jet engine, sudden expansions, and ejectors. In such applications the streams are often parallel so the static pressure can be regarded as uniform, but the velocity varies in the direction normal to the stream.

Complete and incomplete similarity

In Chapters 2 and 3 we considered two instructive and fundamentally different, albeit seemingly analogous, problems. In the problem of very intense, instantaneous and infinitely concentrated flooding considered in Chapter 2, following exactly the basic idea demonstrated in the Introduction for a very intense explosion, we arrived at an idealized statement of infinitely concentrated flooding. Applying to this idealized problem the standard procedure of dimensional analysis presented in Chapter 1 we were able to reveal the self-similarity of the solution, to find the self-similar variables and to obtain the solution in a simple closed form.

Deeper consideration showed, however, that this simplicity is illusory and that in making the assumption of an infinitely concentrated flooding we went, we might say, to the brink of an abyss.We demonstrated this when in Chapter 3 we modified the formulation of the problem, seemingly only slightly, by introducing fluid absorption. It would seem that in the modified formulation the same ideal problem statement would be possible and that all our dimensional reasoning would preserve its validity. However, in proceeding with the modified formulation we arrived at a contradiction. It turned out that in the modified formulation the solution to the ideal problem of very intense, instantaneous and infinitely concentrated flooding does not exist.