Introduction

Dense fluid-particle flows in which the direct particle/particle interactions are a dominant feature encompass a diverse range of industrial and geophysical contexts (Jaeger et al. 1996), including, for example, slurry pipelines (Shook and Roco 1991), fluidized beds (Davidson and Harrison 1971), mining and milling operations, ploughing (Weighardt 1975), abrasive water jet machining, food processing, debris flows (Iverson 1997), avalanches (Hutter 1993), landslides, sediment transport, and earthquake-induced soil liquefaction. In many of these applications, stress is transmitted both by shear stresses in the fluid and by momentum exchange during direct particle/particle interactions. Many of the other chapters in this book analyze flow in which the particle concentration is sufficiently low that the particle-particle momentum exchange is negligible.

In this chapter we address those circumstances, usually at high particle concentrations, in which the direct particle/particle interactions play an important role in determining the flow properties. When those interactions dominate the mechanics, the motions are called granular flows and the flow patterns can be quite different from those of conventional fluids. An example is included as Figure 13.1, which shows the downward flow of sand around a circular cylinder. Note the upstream wake of stagnant material in front of the cylinder and the empty cavity behind it.

Within the domain of granular flows, there are, as we shall see, several very different types of flow distinguished by the fraction of time for which particles are in contact. For most slow flows, the particles are in contact most of the time.

Introduction

In Chapter 2 it was assumed that the particles were rigid and therefore not deformed, fissioned, or otherwise modified by the flow. However, there are many instances in which the particles that comprise the disperse phase are radically modified by the forces imposed by the continuous phase. Sometimes those modifications are radical enough to, in turn, affect the flow of the continuous phase. For example, the shear rates in the continuous phase may be sufficient to cause fission of the particles and this, in turn, may reduce the relative motion and therefore alter the global extent of phase separation in the flow.

The purpose of this chapter is to identify additional phenomena and issues that arise when the translating disperse phase consists of deformable particles, namely bubbles, droplets, or fissionable solid grains.

Deformation due to Translation

Dimensional Analysis

Because the fluid stresses due to translation may deform the bubbles, drops, or deformable solid particles that make up the disperse phase, we should consider not only the parameters governing the deformation but also the consequences in terms of the translation velocity and the shape. We concentrate here on bubbles and drops in which surface tension, S, acts as the force restraining deformation. However, the reader will realize that there would exist a similar analysis for deformable elastic particles. Furthermore, the discussion is limited to the case of steady translation, caused by gravity, g. Clearly the results could be extended to cover translation due to fluid acceleration by using an effective value of g as indicated in Section 2.4.2.

Introduction

In Chapter 9, the analyses were predicated on the existence of an effective barotropic relation for the homogeneous mixture. Indeed, the construction of the sonic speed in Sections 9.3.1 and 9.3.3 assumes that all the phases are in dynamic equilibrium at all times. For example, in the case of bubbles in liquids, it is assumed that the response of the bubbles to the change in pressure, δp, is an essentially instantaneous change in their volume. In practice this would be the case only if the typical frequencies experienced by the bubbles in the flow are very much smaller than the natural frequencies of the bubbles themselves (see Section 4.4.1). Under these circumstances the bubbles would behave quasistatically and the mixture would be barotropic. However, there are a number of important contexts in which the bubbles are not in equilibrium and in which the nonequilibrium effects have important consequences. One example is the response of a bubbly multiphase mixture to high-frequency excitation. Another is a bubbly cavitating flow where the nonequilibrium bubble dynamics lead to shock waves with substantial noise and damage potential.

In this chapter we therefore examine some flows in which the dynamics of the individual bubbles play an important role. These effects are included by incorporating the Rayleigh–Plesset equation (Rayleigh 1917, Knapp et al. 1970, Brennen 1995) into the global conservation equations for the multiphase flow. Consequently the mixture no longer behaves barotropically.

Introduction

One of the most common requirements of a multiphase flow analysis is the prediction of the energy gains and losses as the flow proceeds through the pipes, valves, pumps, and other components that make up an internal flow system. In this chapter we attempt to provide a few insights into the physical processes that influence these energy conversion processes in a multiphase flow. The literature contains a plethora of engineering correlations for pipe friction and some data for other components such as pumps. This chapter provides an overview and some references to illustrative material but does not pretend to survey these empirical methodologies.

As might be expected, frictional losses in straight uniform pipe flows have been the most widely studied of these energy conversion processes and so we begin with a discussion of that subject, focusing first on disperse or nearly disperse flows and then on separated flows. In the last part of the chapter, we consider multiphase flows in pumps, in part because of the ubiquity of these devices and in part because they provide a second example of the multiphase flow effects in internal flows.

Frictional Loss in Disperse Flow

Horizontal Flow

We begin with a discussion of disperse horizontal flow. There exists a substantial body of data relating to the frictional losses or pressure gradient, (−dp/ds), in a straight pipe of circular cross section (the coordinate s is measured along the axis of the pipe).

Introduction

Sprays are an important constituent of many natural and technological processes and range in scale from the very large dimensions of the global air/sea interaction and the dynamics of spillways and plunge pools to the smaller dimensions of fuel injection and ink-jet systems. In this chapter we first examine the processes by which sprays are formed and some of the resulting features of those sprays. Then, because the combustion of liquid fuels in droplet form constitute such an important component of our industrialized society, we focus on the evaporation and combustion of single droplets and follow that with an examination of the features involved in the combustion of sprays.

Types of Spray Formation

In general, sprays are formed when the interface between a liquid and a gas becomes deformed and droplets of liquid are generated. These then migrate out into the body of the gas. Sometimes the gas plays a negligible role in the kinematics and dynamics of the droplet formation process; this simplifies the analyses of the phenomena. In other circumstances the gas dynamic forces generated can play an important role. This tends to occur when the relative velocity between the gas and the liquid becomes large as is the case, for example, with hurricane-generated ocean spray.

Several prototypical flow geometries are characteristic of the natural and technological circumstances in which spray formation is important. The first prototypical geometry is the flow of a gas over a liquid surface.

Introduction

Scope

In the context of this book, the term multiphase flow is used to refer to any fluid flow consisting of more than one phase or component. For brevity and because they are covered in other texts, we exclude those circumstances in which the components are well mixed above the molecular level. Consequently, the flows considered here have some level of phase or component separation at a scale well above the molecular level. This still leaves an enormous spectrum of different multiphase flows. One could classify them according to the state of the different phases or components and therefore refer to gas/solids flows or liquid/solids flows or gas/particle flows or bubbly flows and so on; many texts exist that limit their attention in this way. Some treatises are defined in terms of a specific type of fluid flow and deal with low-Reynolds-number suspension flows, dusty gas dynamics, and so on. Others focus attention on a specific application such as slurry flows, cavitating flows, aerosols, debris flows, fluidized beds, and so on; again, there are many such texts. In this book we attempt to identify the basic fluid mechanical phenomena and to illustrate those phenomena with examples from a broad range of applications and types of flow.

Parenthetically, it is valuable to reflect on the diverse and ubiquitous challenges of multiphase flow. Virtually every processing technology must deal with multiphase flow, from cavitating pumps and turbines to electrophotographic processes to papermaking to the pellet form of almost all raw plastics.

Introduction

This chapter briefly reviews the issues and problems involved in constructing the equations of motion for individual particles, drops, or bubbles moving through a fluid. For convenience we use the generic name particle to refer to the finite pieces of the disperse phase or component. 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 focuses on the construction of such a particle or bubble equation of motion.

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 is denoted by Vi(t). The velocity that the fluid would have had at the location of the particle center in the absence of the particle is denoted by Ui(t).

In this chapter the theory of the generation of oceanwaves by wind will be developed resulting in an expression for the wind-input source function Sin of the action balance equation. As will be seen from the subsequent discussion, this problem has led to many debates and much controversy. There may be several reasons for this. On the one hand, from the theoretical point of view it should be realized that one is dealing with an extremely difficult problem because it involves the modelling of a turbulent airflow over a surface that varies in space and time. Although there has been much progress in understanding turbulence over a flat plate in steady-state conditions, modelling attempts of turbulent flow over (nonlinear) gravity waves are only beginning and, as will be seen, there is still a considerable uncertainty regarding the validity of these models.

On the other hand, from an experimental point of view it should be pointed out that it is not an easy task to measure growth rates of waves by wind. First of all, one cannot simply measure growth rates by studying time series of the surface elevation since the time evolution of ocean waves is determined by a number of processes such as wind input, nonlinear interactions and dissipation. In order to measure the growth of waves by wind one therefore has to make certain assumptions regarding the process that causes wave growth.

In this chapter we study the effects of nonlinearity on the evolution of deep-water gravity waves. Eventually this will result in an expression for the source function for nonlinear wave–wave interactions and dissipation (presumably by white capping), which completes the description of the energy balance equation.

We shall begin with a fairly extensive discussion of nonlinear wave–wave interactions, followed by a brief treatment of dissipation of wave energy by white capping. The latter treatment is only very schematic, however, because this process involves steep waves which only occur sporadically. At best the choice of the white-capping source function can be made plausible. It turns out that the overall dissipation rate is in agreement with observed dissipation rates. Much more is known regarding nonlinear wave–wave interactions. An important reason for this is that ocean waves may be regarded most of the time as weakly nonlinear, dispersive waves. Because of this there is a small parameter present which permits the study of the effect of nonlinearity on wave evolution by means of a perturbation expansion with starting point linear, freely propagating ocean waves. In addition, it should be pointed out that the subject of nonlinear ocean waves has conceptually much in common with nonlinear wave phenomena arising in diverse fields such as optics and plasma physics. In particular, since the beginning of the 1960s many people have contributed to a better understanding of the properties of nonlinear waves, and because of the common denominater we have seen relatively rapid progress in the field of nonlinear ocean waves.

The subject of ocean waves and their generation by wind has fascinated me greatly since I started towork in the Department of Oceanography at the Royal Netherlands Meteorological Institute (KNMI) at the end of 1979. The wind-induced growth of water waves on a pond or a canal is a daily experience for those who live in the lowlands, yet it appeared that this process was hardly understood. Gerbrand Komen, who arrived 2 years earlier at KNMI and who introduced me to this field, pointed out that the most prominent theory explaining wave growth by wind was the Miles (1957) theory which relied on a resonant interaction between wind and waves. Since I did my Ph.D. in plasma physics, I noticed immediately an analogy with the problem of the interaction of plasma waves and electrons; this problem has been studied extensively both experimentally and theoretically. The plasma waves problem has its own history. It was Landau (1946), who discovered that depending on the slope of the particle distribution function at the location where the phase velocity of the plasma wave equals the particle velocity, the plasma wave would either grow or damp. Because of momentum and energy conservation this would result in a modification of the particle-velocity distribution. For a spectrum of growing plasma waves with random phase, this problem was addressed in the beginning of the 1960s by Vedenov et al. (1961) and by Drummond and Pines (1962).

This is a book about ocean waves, their evolution and their interaction with the environment. It presents a summary and unification of my knowledge of wave growth, nonlinear interactions and dissipation of surface gravity waves, and this knowledge is applied to the problem of the two-way interaction of wind and waves, with consequences for atmosphere and ocean circulation.

The material of this book is, apart from my own contributions, based on a number of sources, ranging from the works of Whitham and Phillips to the most recent authoritative overview in the field of ocean waves, namely the work written by the WAM group, Dynamics and Modelling of Ocean Waves. Nevertheless, the present book is limited in its scope because it will hardly address interesting issues such as the assimilation of observations, the interpretation of satellite measurements from, for example, the radar altimeter, the scatterometer and the synthetic-aperture radar, nor will it address shallow-water effects. These are important issues but I felt that the reader would be served more adequately by concentrating on a limited number of subjects, emphasizing the role of ocean waves in practical applications such as wave forecasting and illuminating their role in the air–sea momentum exchange.

I started working on this book some 8 years ago. It would never have been finished had it not been for the continuous support of my wife Danielle Mérelle.

In this chapter we shall try to derive, from first principles, the basic evolution equation for ocean-wave modelling which has become known as the energy balance equation. The starting point is the Navier–Stokes equations for air and water. The problem of wind-generated ocean waves is, however, a formidable one, and several approximations and assumptions are required to arrive at the desired result. Fortunately, there are two small parameters in the problem, namely the steepness of the waves and the ratio of air density to water density. As a result of the relatively small air density, the momentum and energy transfer from air to water is relatively small so that, because of wind input, it will take many wave periods to have an appreciable change of wave energy. In addition, the steepness of the waves is expected to be relatively small. In fact, the assumption of small wave steepness may be justified a posteriori. Hence, because of these two small parameters one may distinguish two scales in the time–space domain, namely a short scale related to the period and wavelength of the ocean waves and a much longer time and length scale related to changes due to small effects of nonlinearity and the wind-induced growth of waves.

Using perturbation methods, an approximate evolution equation for the amplitude and the phase of the deep-water gravity waves may be obtained.

In this book we have given an overview of the role that ocean waves play in the problem of the interaction of atmosphere and ocean. However, in order to appreciate this role we had to elaborate on how ocean waves evolve in space and time. It was found that ocean waves evolve according to the well-known energy balance equation which states that the wave spectrum changes due to advection with the group velocity, and due to physical processes such as generation by wind, nonlinear transfer by four-wave interactions and dissipation by, for example, white capping. A detailed exposition of the derivation of the physical source functions was given, followed by a study of the impact of ocean waves on the atmospheric circulation and one aspect of the ocean circulation, namely storm surges. It was also pointed out that the study of the effects of ocean waves on ocean circulation is only beginning, but that promising improvements on its wind-driven component are expected in the near future.

This book was concluded by an extensive discussion of the verification of the ECMWF forecast of wave parameters, such as significant wave height. The impression from this verification study is that the quality of the ECMWF wave analysis and wave forecast is high, certainly if the results are put in a historical perspective.