Introduction

Generally, the liquid hydrodynamic pressure in rigid containers has two distinct components. One component is directly proportional to the acceleration of the tank and is caused by part of the fluid moving in unison with the tank. The second, known as “convective” pressure, experiences sloshing at the free surface. A realistic representation of the liquid dynamics inside closed containers can be approximated by an equivalent mechanical system. The equivalence is taken in the sense of equal resulting forces and moments acting on the tank wall. By properly accounting for the equivalent mechanical system representation of sloshing, the problem of overall dynamic system behavior can be formulated more simply. For linear planar liquid motion, one can develop equivalent mechanical models in the form of a series of mass-spring dashpot systems or a set of simple pendulums. For nonlinear sloshing phenomena, other equivalent models such as spherical or compound pendulum may be developed to emulate rotational and chaotic sloshing.

Graham (1951) developed an equivalent pendulum to represent the free-surface oscillations of a liquid in a stationary tank. Graham and Rodriguez (1952) introduced another model consisting of a sloshing point mass attached with springs to the tank wall at a specified depth and a fixed rigid mass. Pinson (1964) determined spring constants for liquid propellant in ellipsoidal tanks. Models in the form of mass-spring dashpot systems or a set of simple pendulums were considered by Ewart (1956), Bauer (1960a, 1961c, 1962b), Armstrong and Kachigan (1961), Abramson and Ransleben (1961c), Mooney, et al. (1964b).

When I was a very young boy I was enchanted by airplanes. The very idea that such a machine, with no apparent motions of its own – except, of course, for that tiny rotating thing at the front – could fly through the air was amazing. I could see that birds and insects could all fly with great dexterity, but that was because they could flap their wings, and thus support their weight as well as maneuver. And fish could even “fly” through water by motions of their body. How exciting it was then, to begin to learn something about how objects interact with the fluids surrounding them, and the useful consequences of those flows. That ultimately led, of course, to the broad study of fluid dynamics, with all of its wonderful manifestations.

There is hardly a single aspect of our daily lives, and indeed even of the entire universe in which we live, that is not in some way governed or described by fluid dynamics – from the locomotion of marine animals to the birth and death of distant galaxies. As a major field of technical and scientific knowledge, there are vast bodies of literature devoted to almost every facet of fluid behavior: laminar and turbulent flows, discontinuous (separated) flows, vortex flows, internal waves, free surface waves, compressible fluids and shock waves, multi-phase flows, and many, many others. With such a countless array of fluid phenomenon before us, what then leads to the focus of the present work?

Introduction

The dynamic analysis of a cylindrical shell experiencing elastic deformation that is comparable to its wall thickness cannot be described within the framework of the linear theory. The same is applied if the liquid free-surface amplitude is relatively large. In both cases, nonlinear analysis should be carried out. The presence of nonlinearities may result in nonlinear resonance conditions that cause complex response characteristics. One of the main difficulties in nonlinear problems of shell–liquid systems is that the boundary conditions are essentially nonlinear. This is in addition to the fact that the strain state of an elastic shell and the shape of the liquid free surface are not known a priori. The treatment of the nonlinear interaction of a liquid–shell system is a nonclassical boundary-value problem and relies on mechanics of deformable solids, fluid dynamics, and nonlinear mechanics.

With reference to nonlinear vibrations of cylindrical shells in vacuo, the literature is very rich and reports some controversies regarding the influence of nonlinearities on the shell dynamic behavior. The main results have been reviewed by Vol'mir (1972, 1979), Leissa (1973), Evensen (1974), Kubenko, et al. (1984), Amiro and Prokopenko (1997), and Amabili, et al. (1998b). Some attempts have been made to reconcile the reported discrepancies (see, e.g., Dowell, 1998, Evensen, 1999, and Amabili, et al. 1999c). It is believed that Reissner (1955) made the first attempt to study the influence of large-amplitude vibration for simply supported shells.

Sloshing means any motion of the free liquid surface inside its container. It is caused by any disturbance to partially filled liquid containers. Depending on the type of disturbance and container shape, the free liquid surface can experience different types of motion including simple planar, nonplanar, rotational, irregular beating, symmetric, asymmetric, quasi-periodic and chaotic. When interacting with its elastic container, or its support structure, the free liquid surface can exhibit fascinating types of motion in the form of energy exchange between interacting modes. Modulated free surface occurs when the free-liquid-surface motion interacts with the elastic support structural dynamics in the neighborhood of internal resonance conditions. Under low gravity field, the surface tension is dominant and the liquid may be oriented randomly within the tank depending essentially upon the wetting characteristics of the tank wall.

The basic problem of liquid sloshing involves the estimation of hydrodynamic pressure distribution, forces, moments and natural frequencies of the free-liquid surface. These parameters have a direct effect on the dynamic stability and performance of moving containers.

Generally, the hydrodynamic pressure of liquids in moving rigid containers has two distinct components. One component is directly proportional to the acceleration of the tank. This component is caused by the part of the fluid moving with the same tank velocity. The second is known as “convective” pressure and represents the free-surface-liquid motion. Mechanical models such as mass-spring-dashpot or pendulum systems are usually used to model the sloshing part.

Introduction

The problem of dynamic interaction of liquid sloshing with elastic structures may fall under one of the following categories:

1. Interaction of liquid sloshing dynamics with the container elastic modes in breathing and bending. This type is addressed in this chapter and Chapter 9.

2. Interaction of liquid sloshing dynamics with the supporting elastic structure. This type is treated in Chapter 10.

3. Liquid interaction with immersed elastic structures. This class will not be addressed in this book and the reader may consult Chen (1987), Paidoussis (1998) and Dzyuba and Kubenko (2002).

This chapter presents the linear problem of liquid interaction with its elastic container. Two limiting cases may occur where interaction disappears. The first case deals with the excitation of liquid surface modes where significant elastic modes of the container are not participating. In this case, the analysis of liquid dynamics in a rigid container will provide a satisfactory description of the overall behavior. The second case deals with the excitation of the container elastic modes where significant liquid motion does not occur. In this case, the presence of liquid will contribute to the mass distributed to the tank walls, and the analysis can be carried out without considering any interaction with liquid sloshing dynamics.

The first step in studying the interaction of liquid dynamics with elastic tank dynamics is to consider the linear eigenvalue problem and response to external excitations. The coupling may take place between the liquid-free-surface dynamics and with either the tank bending oscillations or breathing modes (or shell modes).

The subject of multiphase flows encompasses a vast field, a host of different technological contexts, a wide spectrum of different scales, a broad range of engineering disciplines, and a multitude of different analytical approaches. Not surprisingly, the number of books dealing with the subject is voluminous. For the student or researcher in the field of multiphase flow this broad spectrum presents a problem for the experimental or analytical methodologies that might be appropriate for his/her interests can be widely scattered and difficult to find. The aim of the present text is to try to bring much of this fundamental understanding together into one book and to present a unifying approach to the fundamental ideas of multiphase flows. Consequently the book summarizes those fundamental concepts with relevance to a broad spectrum of multiphase flows. It does not pretend to present a comprehensive review of the details of any one multiphase flow or technological context, although reference to books providing such reviews is included where appropriate. This book is targeted at graduate students and researchers at the cutting edge of investigations into the fundamental nature of multiphase flows; it is intended as a reference book for the basic methods used in the treatment of multiphase flows.

I am deeply grateful to all my many friends and fellow researchers in the field of multiphase flows whose ideas fill these pages. I am particularly indebted to my close colleagues Allan Acosta, Ted Wu, Rolf Sabersky, Melany Hunt, Tim Colonius, and the late Milton Plesset, all of whom made my professional life a real pleasure.

Introduction

From a practical engineering point of view one of the major design difficulties in dealing with multiphase flow is that the mass, momentum, and energy transfer rates and processes can be quite sensitive to the geometric distribution or topology of the components within the flow. For example, the geometry may strongly effect the interfacial area available for mass, momentum, or energy exchange between the phases. Moreover, the flow within each phase or component will clearly depend on that geometric distribution. Thus we recognize that there is a complicated two-way coupling between the flow in each of the phases or components and the geometry of the flow (as well as the rates of change of that geometry). The complexity of this two-way coupling presents a major challenge in the study of multiphase flows and there is much that remains to be done before even a superficial understanding is achieved.

An appropriate starting point is a phenomenological description of the geometric distributions or flow patterns that are observed in common multiphase flows. This chapter describes the flow patterns observed in horizontal and vertical pipes and identifies a number of the instabilities that lead to transition from one flow pattern to another.

Topologies of Multiphase Flow

Multiphase Flow Patterns

A particular type of geometric distribution of the components is called a flow pattern or flow regime and many of the names given to these flow patterns (such as annular flow or bubbly flow) are now quite standard.

Introduction

One of the characteristics of multiphase flows with which the engineer has to contend is that they often manifest instabilities that have no equivalent in single-phase flow (see, for example, Boure et al. 1973, Ishii 1982, Gouesbet and Berlemont 1993). Often the result is the occurence of large pressure, flow-rate, or volume-fraction oscillations that, at best, disrupt the expected behavior of the multiphase flow system (and thus decrease the reliability and life of the components, Makay and Szamody 1978) and, at worst, can lead to serious flow stoppage or structural failure (see, for example, NASA 1970, Wade 1974). Moreover, in many systems (such as pump and turbine installations) the trend toward higher rotational speeds and higher power densities increases the severity of the problem because higher flow velocities increase the potential for fluid/structure interaction problems. This chapter focuses on internal flow systems and the multiphase flow instabilities that occur in them.

System Structure

In the discussion and analysis of system stability, we consider that the system has been divided into its components, each identified by its index, k, as shown in Figure 15.1 where each component is represented by a box. The connecting lines do not depict lengths of pipe that are themselves components. Rather, the lines simply show how the components are connected. More specifically they represent specific locations at which the system has been divided up; these points are called the nodes of the system and are denoted by the index, i.

Introduction

The one-dimensional theory of sedimentation was introduced in a classic article by Kynch (1952), and the methods he used have since been expanded to cover a wide range of other multiphase flows. In Chapter 14 we introduced the concept of drift flux models and showed how these can be used to analyze and understand a class of steady flows in which the relative motion between the phases is determined by external forces and the component properties. The present chapter introduces the use of the drift flux method to analyze the formation, propagation, and stability of concentration (or kinematic) waves. For a survey of this material, the reader may wish to consult Wallis (1969).

The general concept of a kinematic wave was first introduced by Lighthill and Whitham (1955) and the reader is referred to Whitham (1974) for a rigorous treatment of the subject. Generically, kinematic waves occur when a functional relation connects the fluid density with the flux of some physically conserved quantity such as mass. In the present context a kinematic (or concentration) wave is a gradient or discontinuity in the volume fraction, α. We refer to such gradients or discontinuities as local structure in the flow; only multiphase flows with a constant and uniform volume fraction are devoid of such structure. Of course, in the absence of any relative motion between the phases or components, the structure is simply convected at the common velocity in the mixture.

Introduction

This chapter addresses the class of compressible flows in which a gaseous continuous phase is seeded with droplets or particles and in which it is necessary to evaluate the relative motion between the disperse and continuous phases for a variety of possible reasons. In many such flows, the motivation is the erosion of the flow boundaries by particles or drops and this is directly related to the relative motion. In other cases, the purpose is to evaluate the change in the performance of the system or device. Still another motivation is the desire to evaluate changes in the instability boundaries caused by the presence of the disperse phase.

Examples include the potential for serious damage to steam turbine blades by impacting water droplets (e.g., Gardner 1963, Smith et al. 1967). In the context of aircraft engines, desert sand storms or clouds of volcanic dust can not only cause serious erosion to the gas turbine compressor (Tabakoff and Hussein 1971, Smialek et al. 1994, Dunn et al. 1996, Tabakoff and Hamed 1986) but can also deleteriously effect the stall margin and cause engine shutdown (Batcho et al. 1987). Other examples include the consequences of seeding the fuel of a solid-propelled rocket with metal particles to enhance its performance. This is a particularly complicated example because the particles may also melt and oxidize in the flow (Shorr and Zaehringer 1967).

In recent years considerable advancements have been made in the numerical models and methods available for the solution of dilute particle-laden flows.

Introduction

Unlike solid particles or liquid droplets, gas/vapor bubbles can grow or collapse in a flow and in doing so manifest a host of phenomena with technological importance. We devote this chapter to the fundamental dynamics of a growing or collapsing bubble in an infinite domain of liquid that is at rest far from the bubble. Although the assumption of spherical symmetry is violated in several important processes, it is necessary to first develop this baseline. The dynamics of clouds of bubbles or of bubbly flows are treated in later chapters.

Bubble Growth and Collapse

Rayleigh–Plesset Equation

Consider a spherical bubble of radius, R(t) (where t is time), in an infinite domain of liquid whose temperature and pressure far from the bubble are T∞ and p∞(t) respectively. The temperature, T∞, is assumed to be a simple constant because temperature gradients are not considered. Conversely, the pressure, p∞(t), is assumed to be a known (and perhaps controlled) input that regulates the growth or collapse of the bubble.

Though compressibility of the liquid can be important in the context of bubble collapse, it will, for the present, be assumed that the liquid density, ρL, is a constant. Furthermore, the dynamic viscosity, µL, is assumed constant and uniform. It will also be assumed that the contents of the bubble are homogeneous and that the temperature, TB(t), and pressure, pB(t), within the bubble are always uniform. These assumptions may not be justified in circumstances that will be identified as the analysis proceeds.