In Chapter 4 we saw several families of methods for locating and advecting the interface. Here, we focus on one of them: the VOF method. Historically, it was used for free surface flows with only one “fluid,” although it is now routinely used for two-fluid flows. In the VOF method the marker function is represented by the fraction of a computational grid cell which is occupied by the fluid assumed to be the reference phase.

A very large number (probably dozens) of VOF methods have been proposed. When we choose the method we try to strike a balance between several qualities: accuracy, simplicity and volume conservation.

Basic properties

The volume fraction or color function C is the discrete version of the characteristic function H; see Equation (4.3). We will be considering only two-phase or free-surface flows, so that the C data represent the fraction of each grid cell occupied by the reference phase. Furthermore, we restrict our analysis to Cartesian grids with square cells of side h = Δx = Δy.

The function C varies between the constant value one in full cells to zero in empty cells, while mixed cells with an intermediate value of C define the transition region where the interface is localized.

Low-order VOF methods do not need to specify the location of the interface in the transition region, but a geometrical interpretation of these methods shows that in two dimensions the interface line in each mixed cell is represented by a segment parallel to one of the two coordinate axes.

When the governing equations are solved on a fixed grid, using one set of equations for the whole flow field, the different fluids must be identified in some way. This is generally done by using a marker function that takes different values in the different fluids. Sometimes a material property, such as the fluid density for incompressible fluids, can serve as a marker function, but here we shall assume that the rôle of the marker function is only to identify the different fluids. As the fluids move, and the boundary between the different fluids changes location, the marker function must be updated. Updating the marker function accurately is both critical for the success of simulations of multiphase flows and also surprisingly difficult. In this chapter we discuss the difficulties with advecting the marker function directly and the various methods that have been developed to overcome these difficulties.

The VOF method is the oldest method to advect a marker function 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 CIP method. Instead of advecting the marker function directly, the boundary between the different fluids can also be tracked using marker points, and the marker function then reconstructed from the location of the interface. Methods using marker points are generally referred to as “front-tracking” methods to distinguish them from “front-capturing” methods, where the marker function is advected directly.

Various applications and natural processes involve large deformations and eventual breakup of liquid jets, layers, and droplets. When liquid masses fragment in a small number of pieces one speaks of breakup. More intense phenomena where, for instance, a liquid jet is broken into seemingly microscopic droplets are called atomization, although the term is somewhat incorrect, since the individual pieces are still far larger than atomic scales.

Nevertheless, atomization is a striking process in which finely divided sprays or droplet clouds are produced. This is often based on the ejection of a high-speed liquid jet from an atomizer nozzle. Many other configurations exist, such as sheets ejected at high speed from diversely shaped nozzles, or colliding with each other. As with many of the multiphase phenomena investigated in this book, atomization offers a rich physical phenomenology which is still poorly understood. Considerable progress has been made in the development of methods for atomization simulations during the last few years, and advances in hardware are making it possible to conduct simulations of unprecedented complexity.

Introduction

There are many important motivations for the study of spray formation, droplet breakup, and atomization. To take a first example from natural phenomena, spray formation atop ocean waves occurs when sufficiently strong winds strip droplets from the crests of the waves. Breaking waves also create bubbles that, when bursting at the surface, create a very fine mist that can rise high into the atmosphere.

The accurate computation of the surface tension is perhaps one of the most critical aspects of any method designed to follow the motion of the boundary between immiscible fluids for a long time. In methods based on the “one-fluid” formulation, where a fixed grid is used to find the motion, surface tension is added as a body force to the discrete version of the Navier–Stokes equations. Finding the surface force depends on whether the fluid interface is tracked by a direct advection of a marker function (VOF) or whether discrete points are used to mark the interface (front tracking). Here, we describe how to find surface tension for both VOF and front-tracking methods. At the end of the chapter we examine the performance of the various methods and the challenges in computing surface tension accurately and robustly.

Computing surface tension from marker functions

A marker function such as the color function C of the VOF method or the level-set function F is a function that indicates (marks) where the interface is. A whole family of methods for surface tension have been developed for the use of marker functions; however, these methods may also be used in connection with front tracking methods. The standard approach is termed the continuous surface force (CSF) method. We also describe a variant that conserves momentum exactly, the continuous surface stress (CSS) method.

Continuous surface force method

For simplicity, we consider first the case where σ is constant.

Instead of advecting a marker function identifying the different fluids directly, as in the VOF method, the boundary between the fluids can be represented by connected marker points that are moved by the fluid. This approach is usually called front tracking, and in Chapter 4 we discussed the basic idea briefly and gave a short historical overview. In this chapter we describe front tracking in more detail.

The use of connected marker points to track the motion of a complex and deforming fluid interface can lead to several different methods, depending on the details of the implementation. Generally, however, front tracking involves the following considerations:

1. (i) The data structure used to describe the front. Although the use of marker particles simplifies many aspects of the advection of a fluid interface, other aspects become more complex. We use front to refer to the complete set of computational objects used to represent the interface. In addition to the marker points, the front often includes information about the connectivity of the points, as well as a description of the physics at the interface. The management of the front can be greatly simplified by the use of the appropriate data structure. There is a fundamental difference in the level of complexity between fronts in two dimensions and in three dimensions and, in general, any data structure can be made to work reasonably efficiently in two dimensions. […]

It is a remarkable aspect of the “unreasonable effectiveness of mathematics in the natural sciences” (Wigner 1960) that a handful of equations are sufficient to describe mathematically a vast number of physically disparate phenomena, at least at some level of approximation. Key reasons are the isotropy and uniformity of space-time (at least locally), the attendant conservation laws, and the useful range of applicability of linear approximations to constitutive relations.

After a very much abbreviated survey of the principal properties of vector fields, we present a summary of these fundamental equations and associated boundary conditions, and then describe several physical contexts in which they arise. The initial chapters of a book on any specific discipline give a far better derivation of the governing equations for that discipline than space constraints permit here. Our purpose is, firstly, to remind the reader of the meaning accorded to the various symbols in any specific application and of the physics that they describe and, secondly, to show the similarity among different phenomena.

The final section of this chapter is a very simple-minded description of the method of eigenfunction expansion systematically used in many of the applications treated in this book. The starting point is an analogy with vectors and matrices in finite-dimensional spaces and the approach is purposely very elementary; a “real” theory is to be found in Part III of the book.

Green's functions permit us to express the solution of a non-homogeneous linear problem in terms of an integral operator of which they are the kernel. We have already presented in simple terms this idea in §2.4. We now give a more detailed theory with applications mainly to ordinary differential equations. The next chapter deals with Green's functions for partial differential equations.

The determination of a Green's function requires the solution of a problem similar to (although somewhat simpler than) the original one, but the effort required is balanced by several advantages. In the first place, and at the most superficial level, once the Green's function G is known, it is unnecessary to solve the problem ex novo for every new set of data: it is sufficient to allow G to act on the new data to have the solution directly. Secondly, and most importantly for our purposes, Green's function theory provides a foundation for the various eigenfunction expansion and integral transform methods used in Part I of this book. Thirdly, even if the Green's function cannot be determined explicitly, one can base on it the powerful boundary integral numerical method outlined in §16.1.3 of the next chapter. Furthermore, once an expression for the solution of a problem – even if not fully explicit – is available, it becomes possible to deduce several important features of it, including bounds existence, uniqueness and others.

In many ways the sphere is the prototypical three-dimensional body and the consideration of fields in the presence of spherical boundaries sheds light on several features of more general three-dimensional cases.

In all the examples of this chapter extensive use is made of expansions in series of Legendre polynomials, for axi-symmetric problems, or spherical harmonics, for the general three-dimensional case. After a review of the polar coordinate system, we begin with a summary of the properies of these functions which are dealt with in greater detail in Chapters 13 and 14, respectively. While the axi-symmetric situation is somewhat simpler, it is also contained as a special case in the general three-dimensional one and it is therefore expedient to treat it as a special case of the latter.

We start with the general solution of the Laplace and Poisson equations (§7.3) and apply it to several axisymmetric (§§7.4 and 7.5) and non-axisymmetric situations. In all these cases the radial part of the solution consists of powers of r. The examples in the second part of the chapter (§7.13 and §7.14) deal with the scalar Helmholtz equation, for which the radial dependence is expressed in terms of spherical Bessel functions, the fundamental properties of which are summarized in §7.12. The last four sections deal with problems involving vector fields and vector harmonics.