Preface

Gas–liquid flows are ubiquitous in industrial and environmental processes. Examples are the transportation of petroleum products, the cooling of nuclear reactors, the operation of absorbers, distillation columns, gas lift pumps. Quite often corrosion and process safety depend on the configuration of the phases. Thus, the interest in this area should not be surprising.

The goal of this book is to give an account of scientific tools needed to understand the behavior of gas–liquid systems and to read the scientific literature. Particular emphasis is given to flow in pipelines.

The following brief historical account is taken from a plenary lecture by the author at the Third International Conference on Multiphase Flow, Lyon, France, June 8–12, 1998. (Int. J. Multiphase Flow 26, 169–190, 2000):

Introduction

The “simplest” models for gas–liquid flow systems are ones for which the velocity is uniform over a cross-section and unidirectional. This includes flows in a long straight pipe and steady flows in a nozzle.

A treatment of pipe flow with a constant cross-section is initiated by reviewing analyses of incompressible and compressible single-phase flows. A simple way to use these results is to describe gas–liquid flows with a homogeneous model that assumes the phases are uniformly distributed, that there is no slip between the phases and that the phases are in thermodynamic equilibrium. The volume fraction of the gas, α, is then directly related to the relative mass flows of the phases. However, the assumption of no slip, S = 1, can introduce considerable error. This has prompted a consideration of a separated flow model, where uniform flows of gas and liquid are pictured as moving parallel to one another with different velocities and to be in thermodynamic equilibrium.

This book is intended as a combination of a reference book for those who work with cavitation or bubble dynamics and as a monograph for advanced students interested in some of the basic problems associated with this category of multiphase flows. A book like this has many roots. It began many years ago when, as a young postdoctoral Fellow at the California Institute of Technology, I was asked to prepare a series of lectures on cavitation for a graduate course cum seminar series. It was truly a baptism by fire, for the audience included three of the great names in cavitation research, Milton Plesset, Allan Acosta, and Theodore Wu, none of whom readily accepted superficial explanations. For that, I am immensely grateful. The course and I survived, and it evolved into one part of a graduate program in multiphase flows.

There are many people to whom I owe a debt of gratitude for the roles they played in making this book possible. It was my great good fortune to have known and studied with six outstanding scholars, Les Woods, George Gadd, Milton Plesset, Allan Acosta, Ted Wu, and Rolf Sabersky. I benefited immensely from their scholarship and their friendship. I also owe much to my many colleagues in the American Society of Mechanical Engineers whose insights fill many of the pages of this monograph. The support of my research program by the Office of Naval Research is also greatly appreciated.

Introduction

When the concentration of bubbles in a flow exceeds some small value the bubbles will begin to have a substantial effect on the fluid dynamics of the suspending liquid. Analyses of the dynamics of this multiphase mixture then become significantly more complicated and important new phenomena may be manifest. In this chapter we discuss some of the analyses and phenomena that may occur in bubbly multiphase flow.

In the larger context of practical multiphase (or multicomponent) flows one finds a wide range of homogeneities, from those consisting of one phase (or component) that is very finely dispersed within the other phase (or component) to those that consist of two separate streams of the two phases (or components). In between are topologies that are less readily defined. The two asymptotic states are conveniently referred to as homogeneous and separated flow. One of the consequences of the topology is the extent to which relative motion between the phases can occur. It is clear that two different streams can readily travel at different velocities, and indeed such relative motion is an implicit part of the study of separated flows. On the other hand, it is clear from the results of Section 5.11 that any two phases could, in theory, be sufficiently well mixed and the disperse particle size sufficiently small so as to eliminate any significant relative motion.

Introduction

This first chapter will focus on the mechanisms of formation of two-phase mixtures of vapor and liquid. Particular attention will be given to the process of the creation of vapor bubbles in a liquid. In doing so we will attempt to meld together several overlapping areas of research activity. First, there are the studies of the fundamental physics of nucleation as epitomized by the books of Frenkel (1955) and Skripov (1974). These deal largely with very pure liquids and clean environments in order to isolate the behavior of pure liquids. On the other hand, most engineering systems are impure or contaminated in ways that have important effects on the process of nucleation. The later part of the chapter will deal with the physics of nucleation in such engineering environments. This engineering knowledge tends to be divided into two somewhat separate fields of interest, cavitation and boiling. A rough but useful way of distinguishing these two processes is to define cavitation as the process of nucleation in a liquid when the pressure falls below the vapor pressure, while boiling is the process of nucleation that ocurs when the temperature is raised above the saturated vapor/liquid temperature. Of course, from a basic physical point of view, there is little difference between the two processes, and we shall attempt to review the two processes of nucleation simultaneously.

Introduction

In the preceding chapter some of the equations of bubble dynamics were developed and applied to problems of bubble growth. In this chapter we continue the discussion of bubble dynamics but switch attention to the dynamics of collapse and, in particular, consider the consequences of the violent collapse of vapor-filled cavitation bubbles.

Bubble Collapse

Bubble collapse is a particularly important subject because of the noise and material damage that can be caused by the high velocities, pressures, and temperatures that may result from that collapse. The analysis of Section 2.4 allowed approximate evaluation of the magnitudes of those velocities, pressures, and temperatures (Equations (2.36), (2.38), (2.39)) under a number of assumptions including that the bubble remains spherical. It will be shown in Section 3.5 that collapsing bubbles do not remain spherical. Moreover, as we shall see in Chapter 7, bubbles that occur in a cavitating flow are often far from spherical. However, it is often argued that the spherical analysis represents the maximum possible consequences of bubble collapse in terms of the pressure, temperature, noise, or damage potential. Departure from sphericity can diffuse the focus of the collapse and reduce the maximum pressures and temperatures that might result.

Introduction

This chapter will briefly review the issues and problems involved in constructing the equations of motion for individual bubbles (or drops or solid particles) moving through a fluid and will therefore focus on the dynamics of relative motion rather than the dynamics of growth and collapse. For convenience we shall use the generic name “particle” when any or all of bubbles, drops, and solid particles are being considered. The analyses are implicitly confined to those circumstances in which the interactions between neighboring particles are negligible. In very dilute multiphase flows in which the particles are very small compared with the global dimensions of the flow and are very far apart compared with the particle size, it is often sufficient to solve for the velocity and pressure, ui(xi, t) and p(xi, t), of the continuous suspending fluid while ignoring the particles or disperse phase. Given this solution one could then solve an equation of motion for the particle to determine its trajectory. This chapter will focus on the construction of such a particle or bubble equation of motion. Interactions between particles or, more particularly, bubble, are left for later.

The body of fluid mechanical literature on the subject of flows around particles or bodies is very large indeed. Here we present a summary that focuses on a spherical particle of radius, R, and employs the following common notation. The components of the translational velocity of the center of the particle will be denoted by Vi(t).

Introduction

Having considered the initial formation of bubbles, we now proceed to identify the subsequent dynamics of bubble growth and collapse. The behavior of a single bubble in an infinite domain of liquid at rest far from the bubble and with uniform temperature far from the bubble will be examined first. This spherically symmetric situation provides a simple case that is amenable to analysis and reveals a number of important phenomena. Complications such as those introduced by the presence of nearby solid boundaries will be discussed in the chapters which follow.

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∞ andp∞(t) respectively. The temperature, T∞, is assumed to be a simple constant since temperature gradients were eliminated a priori and uniform heating of the liquid due to internal heat sources or radiation will not be considered. On the other hand, the pressure, P∞(t), is assumed to be a known (and perhaps controlled) input which 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.

Introduction

We begin this discussion of cavitation in flows by describing the effect of the flow on a single cavitation “event.” This is the term used in referring to the processes that occur when a single cavitation nucleus is convected into a region of low pressure within the flow, grows explosively to macroscopic size, and collapses when it is convected back into a region of higher pressure. Pioneering observations of individual cavitation events were made by Knapp and his associates at the California Institute of Technology in the 1940s (see, for example, Knapp and Hollander 1948) using high-speed movie cameras capable of 20,000 frames per second. Shortly thereafter Plesset (1948), Parkin (1952), and others began to model these observations of the growth and collapse of traveling cavitation bubbles using modifications of Rayleigh's original equation of motion for a spherical bubble. Many analyses and experiments on traveling bubble cavitation followed, and a brief description these is included in the next section. All of the models are based on two assumptions: that the bubbles remain spherical and that events do not interact with one another.

Introduction

The focus of the two preceding chapters was on the dynamics of the growth and collapse of a single bubble experiencing one period of tension. In this chapter we review the response of a bubble to a continuous, oscillating pressure field. Much of the material comes within the scope of acoustic cavitation, a subject with an extensive literature that is reviewed in more detail elsewhere (Flynn 1964; Neppiras 1980; Plesset and Prosperetti 1977; Prosperetti 1982,1984; Crum 1979; Young 1989). We include here a brief summary of the basic phenomena.

One useful classification of the subject uses the magnitude of the bubble radius oscillations in response to the imposed fluctuating pressure field. Three regimes can be identified:

1. For very small pressure amplitudes the response is linear. Section 4.2 contains the first step in any linear analysis, the identification of the natural frequency of an oscillating bubble.

2. Due to the nonlinearities in the governing equations, particularly the Rayleigh-Plesset Equation (2.12), the response of a bubble will begin to be affected by these nonlinearities as the amplitude of oscillation is increased. Nevertheless the bubble may continue to oscillate stably. Such circumstances are referred to as “stable acoustic cavitation” to distinguish them from those of the third regime described below. Several different nonlinear phenomena can affect stable acoustic cavitation in important ways. Among these are the production of subharmonics, the phenomenon of rectified diffusion, and the generation of Bjerknes forces. Each of these is described in greater detail later in the chapter.

3. …

Introduction

In this chapter we briefly survey the extensive literature on fully developed cavity flows and the methods used for their solution. The terms “free streamline flow” or “free surface flow” are used for those situations that involve a “free” surface whose location is initially unknown and must be found as a part of the solution. In the context of some of the multiphase flow literature, they would be referred to as separated flows. In the introduction to Chapter 6 we described the two asymptotic states of a multiphase flow, homogeneous and separated flow. Chapter 6 described some of the homogeneous flow methods and their application to cavitating flows; this chapter presents the other approach. However, we shall not use the term separated flow in this context because of the obvious confusion with the accepted, fluid mechanical use of the term.

Fully developed cavity flows constitute one subset of free surface flows, and this survey is intended to provide information on some of the basic properties of these flows as well as the methods that have been used to generate analytical solutions of them. A number of excellent reviews of free streamline methods can be found in the literature, including those of Birkkoff and Zarantonello (1957), Parkin (1959), Gilbarg (1960), Woods (1961), Gurevich (1961), Sedov (1966), and Wu (1969,1972). Here we shall follow the simple and elegant treatment of Wu (1969,1972).