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.

The influence and importance of numerical models and simulations in science and engineering as appropriate tools for:

• analysis of engineering processes, as well as for

• conception and design of processes, and the

• development and analysis of control mechanisms,

has rapidly increased in recent years. In some technical research and development areas, simulation has been employed as an important contribution, given identical ranking as experiment and theory. This influence is valid not only in universities and research laboratories, but also in industry.

This technical progress of numerical simulation tools is based on ongoing rapid developments that have been achieved in hardware and numerics, and also on some important developments in modelling of physical and technical processes. These models can be incorporated and implemented into simulation codes that become easy to use. In recent developments in this area, similar success compared to experimental or physical measurement techniques have been achieved.

The possibility of using a simulation model to decouple some of the physical effects and mechanisms involved in a complex technical process, which may only be sequentially analysed by experimental means, highlights the potential of this new analytical approach. Here physical understanding of complex processes may be derived and used to optimize and develop processes. This contributes not only to scientific understanding, but also to economic and ecological technical innovations.

As an example of modelling and numerical process simulation, in this book fluid atomization processes and the spray forming of metals have been investigated, with particular reference to transport and exchange processes within multiphase flow, including momentum, heat and mass transfer.

Reciprocity

Reciprocity is a good thing. Something is given and something else, equally or more valuable, is returned. So it is in reciprocity for states of deformation of elastic bodies. What is received in return is the main benefit from the reciprocal relationship. From a known solution to one loading case, some important aspect of, or the complete solution to, another loading case is returned. The return is, however, not always a complete solution, but sometimes an equation for computing such a solution.

For dynamic systems the concept of reciprocity goes back to the nineteenth century. A pertinent reciprocity theorem was first formulated by von Helmholtz (1860). Lord Rayleigh (1873, 1877), subsequently derived a quite general reciprocity relation for the time-harmonic motion of a linear dynamic system with a finite or infinite number of degrees of freedom. Rayleigh's formulation included the effects of dissipation. In a later work Lamb (1888) attributed the following general reciprocity theorem to von Helmholtz (1886):

Introduction

In Chapter 7 it was shown that for time-harmonic wave motion there is an infinite number of wave modes that can propagate in a linearly elastic layer. These modes correspond to standing waves across the thickness of the layer and propagation along the layer. It was shown that for the isotropic case the standing waves consist of thickness-stretch and thickness-shear motions carried by the wave propagating along the layer, the carrier wave. The carrier wave acts like a membrane wave in the mid-plane of the layer in that it is governed by a reduced wave equation in that plane. As was discussed earlier, the carrier wave may be a plane, cylindrical or any other kind of wave as long as it is a solution of the membrane equation. The important point is that the thickness motions remain the same; they are independent of the form of the carrier wave. For a specific real-valued frequency a wavenumber-like quantity is the solution to the Rayleigh–Lamb frequency equations. The curves that represent the solution for the real, imaginary or complex-valued wavenumber versus the frequency define the frequency spectrum. Each line of frequency versus wavenumber is called a branch and defines a mode of wave propagation in the layer. The modes can be separated into symmetric and antisymmetric modes. For a detailed discussion of the frequency spectrum of Lamb waves we refer to Mindlin (1960) or Achenbach (1973).

The modes are independent from each other in that they satisfy orthogonality relations.

Introduction

For practical purposes much interest exists in the analysis of wave motion in elastic bodies that are geometrically defined by one or two large length parameters and at least one small length parameter. Examples are plates, beams and rods. The exact treatment by analytical methods of wave motion in such structural components is often very difficult, if not impossible. For that reason several one- or two-dimensional models that provide approximate descriptions have been developed. These models are based on a priori assumptions with regard to the form of the displacements across the smaller dimension(s) of the component, generally in the cross-sectional area. For beams and rods the assumptions simplify the description of the kinematics to such an extent that the wave motions can be described by one-dimensional approximate theories. For the propagation of time-harmonic waves it was found that the approximate theories can account adequately for the dispersive behavior of at least the lowest mode of the exact solution over a limited but significant range of wavenumbers and frequencies.

One of the best-known examples is the Bernoulli–Euler beam theory. In this simplest model for the description of flexural motions of beams of arbitrary but small uniform cross section with a plane of symmetry, it is assumed that the dominant displacement component is parallel to the plane of symmetry. It is also assumed that the deflections are small and that the cross-sectional area remains plane and normal to the neutral axis.

In this chapter, fundamental features of the metal spray forming process are introduced in terms of their science and applications. Chapter 1 saw the division of the process into three main steps:

1. disintegration (or atomization),

2. spray establishment, and

3. compaction.

Now, a more detailed introduction to those subprocesses that are especially important for application within the spray forming process, will be given.

The spray forming process

Spray forming is a metallurgical process that combines the main advantages of the two classical approaches to base manufacturing of sophisticated materials and preforms, i.e.:

• metal casting: involving high-volume production and near-net shape forming,

• powder metallurgy: involving near-net shape forming (at small volumes) to yield a homogeneous, fine-grained microstructure.

The spray forming process essentially combines atomization and spraying of a metal melt with the consolidation and compaction of the sprayed mass on a substrate. A typical technical plant sketch and systematic scheme of the spray forming process (as realized within several technical facilities and within the pilot-plant-scale facilities at the University of Bremen, which will be mainly referenced here) is illustrated in Figure 2.1. In the context of spray forming, a metallurgically prepared and premixed metal melt is distributed from the melting crucible via a tundish into the atomization area. Here, in most applications, inert gas jets with high kinetic energy impinge onto the metal stream and cause melt disintegration (twin-fluid atomization). In the resulting spray, the droplets are accelerated towards the substrate and thereby cool down and partly solidify due to intensive heat transfer to the cold atomization gas.

Analysis of turbulent multiphase flow in a spray is of major concern during numerical modelling and simulation, as the turbulence is responsible for a number of subprocesses that affect spray forming applications. These result from coupled transport between drop and gaseous phases, and from extensive transfer of momentum, heat and mass between phases due to the huge exchange area of the combined droplet surface. Physical modelling and description of these exchange and transport processes is key to the understanding of spray proces.

In spray forming, especially, the thermal and kinetic states of melt particles at the point of impingement onto the substrate, or the already deposited melt layer, are of importance. This is the main boundary condition for analysis of growth, solidification and cooling processes in spray formed deposits. These process conditions finally determine the product quality of spray deposited preforms. By impinging and partly compacting particles from the spray, a source for heat (enthalpy), momentum and mass for the growing deposit is generated. The main parameters influencing successful spray simulation in this context are:

• the local temperature distribution and local distribution ratio between the particles and the surface of the deposit,

• particle velocities at the point of impingement, and

• the mass and enthalpy fluxes (integrated rates per unit area and time) of the compacting particles.

Distribution of these properties at the point of impingement is determined mainly by the fragmentation process and by the transport and exchange mechanisms in the spray.

Introduction

Fields of classical physics such as electromagnetic wave theory, acoustics and elastodynamics all have their own reciprocity theorems, which for acoustics and elastodynamics have been discussed in the preceding chapters. A comparable discussion of reciprocity in electromagnetic wave theory is outside the scope of this book. Moreover, there are already several books that have dealt in considerable detail with electromagnetic reciprocity; see e.g., Collin (1960), Auld (1973) and de Hoop (1995).

Interesting applications of reciprocity relations for the interactions of electromagnetic and elastodynamic fields to non-destructive evaluation, particularly as it relates to piezoelectricity, have not received the attention that they deserve, with the exception of the work by Auld (1979). In the present chapter we therefore attempt to correct for this lack of exposure by a discussion of reciprocity for piezoelectric systems.

General reciprocity relations involving coupled electromagnetic and elastic waves were first presented by Foldy and Primakoff (1945) and Primakoff and Foldy (1947), who used these relations to demonstrate the interchangeability of source and receiver in electro-acoustic transmission measurements. In the important paper by Auld (1979) these relations were used to analyze elastic wave scattering coefficients from observations at the electrical terminals of the electromechanical transducers employed in performing a non-destructive testing experiment. In Auld's paper, an expression was derived that directly relates the electrical signal received by an ultrasonic transducer to the radiation patterns of the transmitting and receiving transducers and to the modified patterns resulting from scattering from a flaw.