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.

Introduction

An important application of the reciprocity relation is its use to generate integral representations. With the aid of the basic singular elastodynamic solution for an unbounded solid, an integral representation can be derived that provides the displacement field at a point of observation in terms of the displacements and tractions on the boundary of a body. In the limit as the point of observation approaches the boundary, a boundary integral equation is obtained. This equation can be solved numerically for the unknown displacements or tractions. The calculated boundary values are subsequently substituted in the original integral representation to yield the desired field variables at an arbitrary point of observation.

The boundary element method is often used for the numerical solution of boundary integral equations. The advantage of the boundary element method for solving boundary integral equations is that the dimensionality of the problem is reduced by one. Rather than calculations in a two- or three-dimensional discretized space, we have calculations for discretized curves or surfaces. For detailed discussions of boundary element methods in elastodynamics we refer to the review papers by Beskos (1987) and Kobayashi (1987). These papers contain numerous additional references. We also mention the book edited by Banerjee and Kobayashi (1992), and a recent book by Bonnet (1995) that has several sections on dynamic problems.

We start this chapter with an exposition of the basic ideas for the simpler, two-dimensional, case of anti-plane strain in Section 11.2.

Modelling of technical production facilities, plants and processes is an integral part of engineering and process technology development, planning and construction. The successful implementation of modelling tools is strongly related to one's understanding of the physical processes involved. Most important in the context of chemical and process technologies are momentum, heat and mass transfer during production. Projection, or scaling, of the unit operations of a complex production plant or process, from laboratory-scale or pilot-plant-scale to production-scale, based on operational models (in connection with well-known scaling-up problems) as well as abstract planning models, is a traditional but important development tool in process technology and chemical engineering. In a proper modelling approach, important features and the complex coupled behaviour of engineering processes and plants may be simulated from process and safety aspects viewpoints, as well as from economic and ecologic aspects. Model applications, in addition, allow subdivision of complex processes into single steps and enable definition of their interfaces, as well as sequential investigation of the interaction between these processes in a complex plant. From here, realization conditions and optimization potentials of a complex process or facility may be evaluated and tested. These days, in addition to classical modelling methods, increased input from mathematical models and numerical simulations based on computer tools and programs is to be found in engineering practice. The increasing importance of these techniques is reflected by their incorporation into educational programmes at universities within mechanical and chemical engineering courses.

Introduction

As discussed in Chapter 9, the modes of wave propagation in an elastic layer are well known from Lamb's (1917) classical work. The Rayleigh–Lamb frequency equations, as well as the corresponding equations for horizontally polarized wave modes, have been analyzed in considerable detail; see Achenbach (1973) and Mindlin (1960). It appears, however, that a simple direct way of expressing wave fields due to the time-harmonic loading of a layer in terms of mode expansions, and a suitable method to obtain the coefficients in the expansions by reciprocity considerations, has so far not been recognized. Of course, wave modes have entered the solutions to problems of the forced wave motion of an elastic layer, at least in the case of surface forces applied normally to the faces of the layer, but via the more cumbersome method of integral transform techniques and the subsequent evaluation of Fourier integrals by contour integration and residue calculus. For examples, we refer to the work of Lyon (1955) for the plane-strain case, and that of Vasudevan and Mal (1985) for axial symmetry.

In this chapter the displacements excited by a time-harmonic point load of arbitrary direction, either applied internally or to one of the surfaces of the layer, are obtained directly as summations over symmetric and/or antisymmetric modes of wave propagation along the layer. This is possible by virtue of an application of the reciprocity relation between time-harmonic elastodynamic states.