Geometric wave theory is the natural extension of WKB theory to situations in which the still layer depth H (and therefore the wave speed) is a slowly varying function of both x and y, and possibly even of t, although we will not consider that case here. In fact, even for constant H geometric wave theory is useful because it allows the computation of the structure of normal modes in bounded domains with irregular shapes, i.e., shapes for which there is no simple explicit expression for the normal modes.

The basic assumption of geometric wave theory is that there is a scale separation between the rapidly varying phase of the wavetrain on the one hand, and the slowly varying layer depth and wavetrain parameters such as amplitude and wavenumber on the other. Of course, in bounded domains the domain size must also be large compared to the wavelength. This basic assumption leads to a flexible and generic asymptotic procedure for solving for the wave field. Eventually, with the inclusion of dispersive effects, geometric wave theory becomes the ray-tracing method, which is the Swiss army knife for computing the asymptotic behaviour of small-scale waves in many fields of physics, including GFD.

A peculiarity of the progression from one-dimensional WKB theory to twodimensional geometric wave theory and finally to dispersive ray tracing is that the structure of the theory becomes easier, not harder,asits generality increases.

Scope

This chapter describes the stratified pattern observed in gas–liquid flows, for which liquid flows along the bottom of a conduit and gas flows along the top. The gas exerts a shear stress on the surface of the liquid. It is desired to calculate the average height of the liquid layer and the pressure gradient for given liquid and gas flow rates. The flow is considered to be fully developed so that the height of the liquid is not changing in the flow direction and the pressure gradient is the same in the gas and liquid flows.

In order to consider stratified flow in circular pipes, the simplified model of the flow pattern, presented by Govier & Aziz (1972), is exploited. The interface is pictured to be flat. At large gas velocities, some of the liquid can be entrained in the gas. This pattern is considered in Section 12.5 entitled “the pool model” for horizontal annular flow.

Prologue

Horizontal annular flows differ from vertical annular flows in that gravity causes asymmetric distributions of the liquid in the wall layer and of droplets in the gas flow. The understanding of this behavior is a central problem in describing this system. Because of these asymmetries, entrainment can increase much more strongly with increasing gas velocity than is found for vertical flows.

Theoretical analyses of the influence of gravity on the distribution of liquid in the wall film and on the distribution of droplets in the gas phase are reviewed. As with vertical annular flows, entrainment is considered to be a balance between the rate of atomization of the wall film and rate of deposition of droplets. Because of the asymmetric film distribution, the local rate of atomization varies around the pipe circumference. This is treated theoretically by assuming that the local rate is the same as would be observed for vertical annular flow. Gravitational settling contributes directly to deposition so that the rate of deposition is enhanced. Thus, at low gas velocities, entrainment can be much smaller for horizontal annular flows than for vertical annular flows.

Prologue

Particles entrained in a turbulent fluid are dispersed by velocity fluctuations; they assume a motion that is related to the fluid turbulence. If the suspension flows through a conduit, deposition on a wall depends on the particle turbulence. An understanding of these processes is needed to describe the annular flow regime for which liquid flows along the walls and as drops in the gas flow. The fraction of liquid that is entrained by the gas depends on the rate at which the film is atomized and the rate at which drops deposit on the film.

Equations for trajectories of spherical drops and bubbles in a turbulent flow field are developed. These are used to relate the turbulence properties and the dispersion of particles to the turbulence properties of the fluid in which they are entrained. Of particular interest is the development of relations for the influence of drop size on drop turbulence and on drop dispersion.

Prologue

A central issue to be addressed in analyzing the behavior of bubbles in a gas–liquid flow is understanding the free-fall velocity of a spherical solid particle and the rise velocity of a spherical bubble in an infinite stationary fluid. Analytical solutions for these systems are available for very low particle Reynolds numbers (Stokes law and the Hadamard equation). A derivation of Stokes law is presented in the first part of this chapter.

Experiments show that Stokes law is valid for particle Reynolds numbers less than unity. For larger ReP, empirical correlations of the drag coefficient are used. The description of the rise velocity of bubbles is complicated by possible contamination of the interface. Measurements of the rise velocity of single bubbles are usually presented as plots of US versus the bubble size for a given system. The structure of these plots reflects changes in the shape and behavior of the bubbles. Very large bubbles take the shape of a cap. A prediction of the rise velocity of these cap bubbles, developed by Batchelor, is presented in Section 8.7.

Need for a phenomenological understanding

The one-dimensional analysis and the correlations for frictional pressure drop and void fraction (presented in Chapter 1) have been widely used as a starting point for engineering designs. However, these correlations have the handicap that the structure of the phase boundaries is ignored. As a consequence, they often give results which are only a rough approximation and overlook phenomena which could be of first-order importance in understanding the behavior of a system.

It is now recognized that the central issue in developing a scientific approach to gas–liquid flows is the understanding of how the phases are distributed and of how the behavior of a multiphase system is related to this structure (Hanratty et al., 2003). Of particular interest is the finding that macroscopic behavior is dependent on small-scale interactions. An example of this dependence is that the presence of small amounts of high molecular weight polymers can change an annular flow into a stratified flow by damping interfacial waves (Al-Sarkhi & Hanratty, 2001a).

Prologue

Chapter 2 gives considerable attention to slug flow because of its central role in understanding the configuration of the phases in horizontal and inclined pipes. Several criteria have been identified to define the boundaries of this regime: (1) viscous large-wavelength instability of a stratified flow; (2) Kelvin–Helmholtz instability of a stratified flow; (3) stability of a slug; (4) coalescence of large-amplitude waves. Bontozoglou & Hanratty (1990) suggested that a sub-critical non-linear Kelvin–Helmholtz instability could be an effective mechanism in pipes with very large diameters, but this analysis has not been tested. A consideration of the stability of a slug emerges as being particularly important. It explains the initiation of slugs for very viscous liquids, for high-density gases, for gas velocities where wave coalescence is important and for the evolution of pseudo-slugs into slugs. Chapter 2 (Section 2.2.5) outlines an analysis of slug stability which points out the importance of understanding the rate at which slugs shed liquid. Section 9.2 continues this discussion by developing a relation for Qsh and for the critical height of the liquid layer needed to support a stable slug. Section 9.3 develops a tentative model for horizontal slug flow. Section 9.4 considers the frequency of slugging.

Necessary conditions for the existence of slugs

Figure 9.1 presents simplified sketches of the front and the tail of a slug in a pipeline. The front has a velocity cF; the back has a velocity cB. The stratified liquid layer in front of the slug has a velocity and area designated by uL1, AL1. The mean velocity of the liquid in the slug is uL3. The slug is usually aerated; the mean volume fraction of gas in the slug is designated by α. The gas at station 1 is moving from left to right at a velocity uG1. The assumption is made that the velocity fields can be approximated as being uniform.