As already mentioned in Chapter 2 dedicated to definitions of wave properties and phenomena related to wave breaking, the breaking probability, or as it is also often called breaking rate or frequency of breaking occurrence is one of the most important statistical characteristics of wave fields that contain the breaking events. Technical definitions for the breaking probability are given in Section 2.5.

Together with the breaking severity (Section 2.7, Chapter 6), the probability defines the wave-energy dissipation due to wave breaking. Knowledge of such dissipation is required across a broad range of wave-related applications, with the wave forecast being the most frequent and obvious, and therefore the breaking occurrence has enjoyed key attention within wave-breaking studies.

Experimental and statistical techniques of breaking-probability studies have been discussed in detail in Chapter 3, and theoretical approaches in Chapter 4. As described in these chapters, in the past parameterisations of the breaking rates in terms of environmental characteristics have usually relied on wind speed or its derivatives. In this book, we have argued that, although the wind is of course essentially responsible for the formation of fields of wind-generated waves, its capacity to directly trigger or even affect a breaking event is only marginal, except perhaps for very strong wind forcing. Breaking mainly happens due to hydrodynamic phenomena, that is due to processes in the wave train itself.

The previous chapter was dedicated to experimental methods of detecting wave breaking, quantifying the breaking probability and severity, and measuring effects related to the breaking, including the wave energy dissipation. In the next chapter, it is logical to describe theoretical methods of describing wave-breaking physics or phenomena leading to the breaking.

While experimental oceanography has produced an abundant variety of techniques and approaches to detect and measure breaking, the theories capable of dealing with wave breaking are few. These should not be confused with analytical methods intended to detect the breaking events in surface-wave records (Section 3.7) or with the statistical methods of quantifying the breaking probability and strength (Section 3.8). Both such groups of analytical techniques are placed into experimental Chapter 3 for a good reason – they principally rely on empirical criteria.

Another significant group of analytical approaches, dealing with the dissipation due to breaking, rather than with the breaking as such is also not included in this chapter. Some of these models are based on assumptions intended to interpret pre-breaking or post-breaking properties of the waves, rather than on working with the physics leading to breaking or driving the breaking and its consequences. Others attempt to deduce the dissipation from differences between wave-evolution predictions done by means of kinetic and dynamic equations. In any case, these are indirect techniques that do not depict the wave-breaking event explicitly. They will be described in Section 7.1.

In previous chapters, we have considered in some detail dissipation of wave energy in the course of individual wave-breaking events. This makes clear physical sense as breaking is an intermittent rather than continuous process, and the breaking rates in realistic circumstances are of the order of a few percent. That is, only a few waves out of a hundred break at any given spot or any given time, and that is sufficient to keep the energy balance in a wave system that experiences a consistent and uninterrupted energy input from the wind.

As discussed in Chapter 6 and throughout the book, it is often the case that more than one wave is affected in a single breaking occurrence. The dynamics of these waves is locked and coupled in many ways, and as far as the wave-energy dissipation is concerned, these energy losses are impossible to separate. Therefore, it usually makes sense to look at the properties of the wave group where the breaking took place rather than investigate the dynamics of some individual wave.

One way or another, there are no theoretical or even experimental approaches that would allow us to describe the breaking-dissipation process as such, in terms of some decay rate, as, for example, gradual energy decline due to the action of viscosity in fluid flows. First of all, unlike most if not all other dissipation causes, breaking dissipation has a start and an end.

Wind-generated waves are the most prominent feature of the ocean surface. As much as the oceans cover a major part of our planet, the waves cover all of the oceans. If there is any object in oceanography that does not need too much of a general introduction, it is the surface waves generated by the wind.

Being such a conspicuous entity, these waves, however, represent one of the most complex physical phenomena of nature. Three major processes are responsible for wave evolution in general, with many more whose significance varies depending on conditions (such as wave-bottom interaction which is only noticeable in shallow areas). The first process is energy and momentum input from the wind. The waves are generated by turbulent wind, and the turbulence is most important both for their initial creation and for subsequent growth (e.g. Miles', 1957; Miles, 1959, 1960; Phillips', 1957; Janssen, 1994, 2004; Belcher & Hunt, 1993; Belcher & Hunt, 1998; Kudryavtsev et al., 2001, among many others). There is, however, no fixed theory of turbulence to begin with. Experimentalists have to deal with tiny turbulent fluctuations of air which are of the order of 10-5-10-6 of the mean atmospheric pressure and which must be measured very close to the water surface, typically below the wave crests (e.g. Donelan et al., 2005).

Direct numerical simulations of multiphase flows are a rapidly growing field. In addition to continuous development of new and better numerical techniques and more extensive studies of the problems discussed so far in this Book, researchers are increasingly looking at new and more complex physical problems. In this chapter we examine briefly a few such extensions. We do not attempt to give an exhaustive list of all new applications of methods based on the “one-fluid” formulation of the fluid equations, but we hope that this introduction to the literature will be useful for our readers.

Additional fields and surface physics

Broadly speaking, new physics consists of new field equations, new surface effects, or both. Adding a new field is the simplest extension, but new fields often add new time- and length-scales. Mass transfer in liquids, for example, can lead to boundary layers that are much thinner than those resulting from either heat transfer or fluid motion. Resolving these boundary layers may introduce much more stringent resolution requirements than those necessary for the same problem in the absence of mass transfer.

The simplest new physics is probably heat transfer, where an advection/diffusion equation is solved for the temperature, and many authors have already studied multifluid problems involving heat transfer. The main complication is that large variations in the thermal conductivity across an interface can require fine grids and often it is better to use the harmonic mean of the conductivities at grid points where it is not defined, in the same way as for large differences in the viscosity (see Section 3.4).

The equations governing multiphase flows, where a sharp interface separates immiscible fluids or phases, are presented in this chapter. We first derive the equations for flows without interfaces, in a relatively standard manner. Then we discuss the mathematical representation of a moving interface and the appropriate jump conditions needed to couple the equations across the interfaces. Finally, we introduce the so-called “one-fluid” approach, where the interface is introduced as a singular distribution in equations written for the whole flow field. The “one-fluid” form of the equations plays a fundamental rôle for the numerical methods discussed in the rest of the book.

General principles

The derivation of the governing equations is based on three general principles: the continuum hypothesis, the hypothesis of sharp interfaces, and the neglect of intermolecular forces. The assumption that fluids can be treated as a continuum is usually an excellent approximation. Real fluids are, of course, made of atoms or molecules. To understand the continuum hypothesis, consider the density or amount of mass per unit volume. If this amount were measured in a box of sufficiently small dimensions ℓ, it would be a wildly fluctuating quantity (see Batchelor (1970), for a detailed discussion). However, as the box side ℓ increases, the density becomes ever smoother, until it is well approximated by a smooth function ρ. For liquids in ambient conditions this happens for ℓ above a few tens of nanometers (1 nm = 10−9 m).

Gas–liquid multiphase flows play an essential role in the workings of Nature and the enterprises of mankind. Our everyday encounter with liquids is nearly always at a free surface, such as when drinking, washing, rinsing, and cooking. Similarly, such flows are in abundance in industrial applications: heat transfer by boiling is the preferred mode in both conventional and nuclear power plants, and bubble driven circulation systems are used in metal processing operations such as steel making, ladle metallurgy, and the secondary refining of aluminum and copper. A significant fraction of the energy needs of mankind is met by burning liquid fuel, and a liquid must evaporate before it burns. In almost all cases the liquid is therefore atomized to generate a large number of small droplets and, hence, a large surface area. Indeed, except for drag (including pressure drops in pipes) and mixing of gaseous fuels, we would not be far off to assert that nearly all industrial applications of fluids involve a multiphase flow of one sort or another. Sometimes, one of the phases is a solid, such as in slurries and fluidized beds, but in a large number of applications one phase is a liquid and the other is a gas. Of natural gas–liquid multiphase flows, rain is perhaps the experience that first comes to mind, but bubbles and droplets play a major role in the exchange of heat and mass between the oceans and the atmosphere and in volcanic explosions.

The one-field formulation of the Navier–Stokes equations described in Chapter 2, where a single set of equations is used to describe the motion of all the fluids present, allows us to use numerical methods developed for single-phase flows. There are, however, two complications: the material properties (usually density and viscosity) generally vary from one fluid to the other and to set these properties we must construct an indicator function that identifies each fluid. We must usually also find the surface tension at the interface. The advection of the indicator function is the topic of Chapters 4 to 6 and finding the surface tension will be dealt with in Chapter 7. In this chapter we discuss numerical methods to solve the Navier–Stokes equations, allowing for variable density and viscosity. We will use the finite-volume method and limit the presentation to regular Cartesian grids. Since the multiphase flows considered in this book all involve relatively low velocities, we will assume incompressible flows.

For any numerical solution of the time-dependent Navier–Stokes equations it is necessary to decide:

1. (i) how the grid points, where the various discrete approximations are stored, are arranged;

2. (ii) how the velocity field is integrated in time;

3. (iii) how the advection and the viscous terms are discretized;

4. (iv) how the pressure equation, resulting from the incompressibility condition, is solved; and

5. (v) how boundary conditions are implemented.

These tasks can be accomplished in a variety of ways, but the approach outlined here has been widely used for multiphase flow simulations and results in a reasonably accurate and robust numerical method.

Progress is usually a sequence of events where advances in one field open up new opportunities in another, which in turn makes it possible to push yet another field forward, and so on. Thus, the development of fast and powerful computers has led to the development of new numerical methods for direct numerical simulations (DNS) of multiphase flows that have produced detailed studies and improved knowledge of multiphase flows. While the origin of DNS of multiphase flows goes back to the beginning of computational fluid dynamics in the early sixties, it is only in the last decade and a half that the field has taken off. We, the authors of this book, have had the privilege of being among the pioneers in the development of these methods and among the first researchers to apply DNS to study relatively complex multiphase flows. We have also had the opportunity to follow the progress of others closely, as participants in numerous meetings, as visitors to many laboratories, and as editors of scientific journals such as the Journal of Computational Physics and the International Journal of Multiphase Flows. To us, the state of the art can be summarized by two observations:

• Even though there are superficial differences between the various approaches being pursued for DNS of multiphase flows, the similarities and commonalities of the approaches are considerably greater than the differences. […]

Introduction

Understanding and predicting bubbly flows is of critical importance in a large number of industrial applications, including boiling heat transfer in power plants, various metallurgical processes, and in bubble columns in the chemical industry. In bubble columns, used for partial oxidation of ethylene to acetaldehyde, isobutene separation, wet oxidation of heavily polluted effluent, and the production of synthetic fuels, for example, gas is injected at the bottom and, as the bubbles rise, the gas diffuses into the liquid and reacts (Furusaki et al., 2001). Bubble columns ranging from tens to hundreds of cubic meters are common in the chemical industry and up to thousands of cubic meters in bioreactors, where longer process times are needed. The absence of any moving parts and their relatively simple construction makes bubble columns particularly attractive for large-scale operations (Deckwer, 1992). Their operation, however, is usually dependent on the size of the vessel and the difficulty of scaling up small pilot models makes numerical predictions important. Similar considerations apply to other bubble systems.

Computational modeling of industrial-size multiphase flow systems must by necessity rely on models of the average flow. Such models range from simple mixture models to more sophisticated two-fluid models, where separate equations are solved for the dispersed and the continuous phase. Since no attempt is made to resolve the unsteady motion of individual bubbles, closure relations are necessary for the unresolved motion and the forces between the bubbles and the liquid.

Droplet collisions and impacts are so spectacular that they have come to symbolize the beauty and fascination of fluid mechanics. Although simulations of two dimensional and axisymmetric systems go back to the early times of two-phase flow simulation, those of fully three-dimensional configurations have become possible only recently. It remains difficult, however, to perform realistic simulations of laboratory experiments.

Introduction

Droplet impacts are of major industrial interest. In what is perhaps the most significant application, fuel droplets impact on the walls of pipes and combustion chambers. There they may spread and form thin films or shatter into a spray of smaller droplets. Impacts also have an obvious relevance to ink-jet printing and spray coating. In other industrial processes, droplet impacts are of interest in metallurgy (Liow et al., 1996; Bierbrauer, 1995) and gas-injection processes. High speed droplet impacts may damage turbines operating with multiphase flows. In hypothetical severe nuclear reactor accidents, molten-core debris may impact on containment walls, splashing at very large velocity. In agriculture, impacts are related to the effect of rain on soil erosion (Farmer, 1973), or the spread of pesticides as they are sprayed on plants. Rain also influences air–sea interactions, enhancing the gas exchange and perhaps damping sea waves (Sainsbury and Cheeseman, 1950; Tsimplis and Thorpe, 1989). Droplet breakup, atomization, impacts, and splashes also cause the accumulation of charge in droplets, as shown by the 1905 Nobel physics laureate Philip Lenard following the work of Hertz (Lenard, 1892).