In the first chapters of this book we have seen methods suitable for a first-principles simulation of the interaction between a fluid and solid objects immersed in it. The associated computational burden is considerable and it is evident that those methods cannot handle large numbers of particles. In this chapter we develop an alternative approach which, while approximate, permits the simulation of thousands, or even millions, of particles immersed in a flow. The key feature which renders this possible is that the exchanges of momentum (and also possibly mass and energy) between the particle and the surrounding fluid are modeled, rather than directly resolved. This implies an approximate representation that is based on incorporating assumptions into the development of the mathematical model.

One of the most common approaches used today to model many particle-laden flows is based on the “point-particle approximation,” i.e. the treatment of individual particles as mathematical point sources of mass, momentum, and energy. This approximation requires an examination of the assumptions and limitations inherent to this approach, aspects that are given consideration in this chapter. Point-particle methods have relatively wide application and have proven a useful tool for modeling many complex systems, especially those comprised of a very large ensemble of particles. Details of the numerical aspects inherent to point-particle treatments are highlighted.

We start by putting point-particle methods into the context established earlier in this text and, in particular, in the previous chapter.

Radiation in the Atmosphere is the third volume in the series A Course in Theoretical Meteorology. The first two volumes entitled Dynamics of the Atmosphere and Thermodynamics of the Atmosphere were first published in the years 2003 and 2004.

The present textbook is written for graduate students and researchers in the field of meteorology and related sciences. Radiative transfer theory has reached a high point of development and is still a vastly expanding subject. Kourganoff (1952) in the postscript of his well-known book on radiative transfer speaks of the three olympians named completeness, up-to-date-ness and clarity. We have not made any attempt to be complete, but we have tried to be reasonably up-to-date, if this is possible at all with the many articles on radiative transfer appearing in various monthly journals. Moreover, we have tried very hard to present a coherent and consistent development of radiative transfer theory as it applies to the atmosphere. We have given principle allegiance to the olympian clarity and sincerely hope that we have succeeded.

In the selection of topics we have resisted temptation to include various additional themes which traditionally belong to the fields of physical meteorology and physical climatology. Had we included these topics, our book, indeed, would be very bulky, and furthermore, we would not have been able to cover these subjects in the required depth.

Boundary integral methods are powerful numerical techniques for solving multiphase hydrodynamic and aerodynamic problems in conditions where the Stokes or potential-flow approximations are applicable. Stokes flows correspond to the low Reynolds number limit, and potential flows to the high Reynolds number regime where fluid vorticity can be neglected. For both Stokes and potential flows, the velocity field in the system satisfies linear governing equations. The total flow can thus be represented as a superposition of flows produced by appropriate point sources and dipoles at the fluid interfaces.

In the boundary integral approach the flow equations are solved directly for the velocity field at the fluid interfaces, rather than in the bulk fluid. Thus, these methods are well suited for describing multiphase systems. Examples of systems for which boundary integral algorithms are especially useful include suspensions of rigid particles or deformable drops under Stokes-flow conditions. Applications of boundary integral methods in fluid dynamics, however, cover a broader range. At one end of this range are investigations of the hydrodynamic mobility of macromolecules; at the other end are calculations of the flow field around an airplane wing in a potential flow approximation. Here we will not address the potential flow case, limiting ourselves to Stokes flow.

Introduction

In the present chapter we discuss boundary integral methods for multiphase flows in the Stokes-flow regime. We review the governing differential equations, derive their integral form, and show how to use the resulting boundary integral equations to determine the motion of particles and drops. Specific issues that are relevant for the numerical implementation of these equations are also described.

The previous chapters have been devoted to methods capable of delivering “numerically exact” solutions of the Navier–Stokes equations as applied to various multiphase flow problems. In spite of their efficiency, these methods still require a substantial amount of computation even for relatively simple cases. It is therefore evident that the simulation of more complex flows approaching those encountered in most natural situations or technological contexts (sediment transport, fluidized beds, electric power generation, and many others) cannot be pursued by those means but must be based on a different approach. Furthermore, even if we did have detailed knowledge, e.g., of the motion of all the particles and of the interstitial fluid, most often, for practical purposes, we would be interested in quantities obtained by applying some sort of averaging to this immense amount of information. This observation suggests that it might be advantageous to attempt to formulate equations governing the time evolution of these averages directly. In this approach, rather than aiming at a detailed solution of the Navier–Stokes equations, we would be satisfied with a reduced description based on simplified mathematical models. While one may try to base such models on intuition, a more reliable way is perhaps to start from the exact equations and carry out a process of averaging which would filter out the inessential details retaining the basic physical processes which determine the behavior of the system.

Introduction

The issue of averaging in multiphase flow is a long-standing one with a history which stretches nearly as far back as for single-phase turbulence.

The principle of invariance in the original form was stated by Ambartsumian (1942) expressing the invariance of the diffusely reflected radiation emerging from a semi-infinite atmosphere to the addition or subtraction of an infinitely thin atmospheric layer. Chandrasekhar (1960) advanced the original from and stated four general principles of invariance which apply to finite atmospheric layers. These principles are not based on the radiative transfer equation, but they are of equal physical validity. We accept Goody's (1964a) statement that the principles of invariance may be viewed as a series of common-sense relations between the scattering and transmission functions with the radiances emerging from the upper and lower boundaries of an atmospheric layer and at some intermediate variable level.

Definitions of the scattering and transmission functions

Let us consider a plane–parallel atmospheric layer of vertical optical thickness τ1 bounded on both sides by a vacuum, see Figure 3.1. The upper boundary of this layer is illuminated by a beam of parallel downward directed radiation S0, while at τ = τ1 no radiation is incident in the upward direction. For simplicity, only short-wave radiation will be considered. However, inclusion of infrared radiation causes no particular difficulties. We call this situation the restricted or standard problem.

This book deals with multiphase flows, i.e. systems in which different fluid phases, or fluid and solid phases, are simultaneously present. The fluids may be different phases of the same substance, such as a liquid and its vapor, or different substances, such as a liquid and a permanent gas, or two liquids. In fluid-solid systems, the fluid may be a gas or a liquid, or gases, liquids, and solids may all coexist in the flow domain.

Without further specification, nearly all of fluid mechanics would be included in the previous paragraph. For example, a fluid flowing in a duct would be an instance of a fluid-solid system. The age-old problem of the fluid-dynamic force on a body (e.g. a leaf in the wind) would be another such instance, while the action of wind on ocean waves would be a situation involving a gas and a liquid.

In the sense in which the term is normally understood, however, multiphase flow denotes a subset of this very large class of problems. A precise definition is difficult to formulate as, often, whether a certain situation should be considered as a multiphase flow problem depends more on the point of view – or even the motivation – of the investigator than on its intrinsic nature. For example, wind waves would not fall under the purview of multiphase flow, even though some of the physical processes responsible for their behavior may be quite similar to those affecting gas–liquid stratified flows, e.g. in a pipe – a prime example of a multiphase system. The wall of a duct or a tree leaf may be considered as boundaries of the flow domain of interest, which would not qualify these as multiphase flow problems.

Nearly half a century of computational fluid dynamics has shown that it is very hard to beat uniform structured grids in terms of ease of implementation and computational efficiency. It is therefore not surprising that a large fraction of the most popular methods for finite Reynolds number multiphase flows today are methods where the governing equations are solved on such grids. The possibility of writing one set of governing equations for the whole flow field, frequently referred to as the “one-fluid” formulation, has been known since the beginning of large-scale computational studies of multiphase flows. It was, in particular, used by researchers at the Los Alamos National Laboratory in the early 1960s for the marker-and-cell (MAC) method, which permitted the first successful simulation of the finite Reynolds number motion of free surfaces and fluid interfaces. This approach was based on using marker particles distributed uniformly in each fluid to identify the different fluids. The material properties were reconstructed from the marker particles and sometimes separate surface markers were also introduced to facilitate the computation of the surface tension. While the historical importance of the MAC method for multiphase flow simulations cannot be overstated, it is now obsolete. In current usage, the term “MAC method” usually refers to a projection method using a staggered grid.

When the governing equations are solved on a fixed grid, the different fluids must be identified by a marker function that is advected by the flow. Several methods have been developed for that purpose. The volume-of-fluid (VOF) method is the oldest and, after many improvements and innovations, continues to be widely used. Other marker function methods include the level-set method, the phase-field method, and the constrained interpolated propagation (CIP) method.

In the previous chapter we have presented segregated solution methods for multifluid models. When the interaction among the phases is very strong, or the processes to be simulated have short time scales, the methods that we now describe, in which the equations are more tightly coupled in the solution procedure, are preferable.

Work on methods of this type received a strong impulse with the development of nuclear reactor thermohydraulic safety codes in the 1970s and 1980s. This activity led to well-known codes such as RELAP, TRAC, SIMMER, and several others. The latest developments of these codes focus on the refinement of models, the inclusion of three-dimensional capabilities, better data structure, and vectorization, rather than fundamental changes in the basic algorithms. By and large, the numerical methods they employ are an outgrowth of the ICE approach (Implicit Continuous Eulerian) developed by Harlow and Amsden (1971) in the late 1960s. While very robust and stable, these methods, described in Section 11.2, are only first-order accurate in space and time and have other shortcomings. The more recent work, some of which is outlined in the second part of this chapter, is based on newer developments in computational fluid dynamics which are summarized in Section 11.3.

A tendency toward more strongly coupled solution methods is also evident in contemporary work springing from the segregated approach described in the previous chapter (see, e.g. Kunz et al., 1998, 1999, 2000). These developments lead to a gradual blurring of the distinction between the two approaches.