7

Flow Patterns

7.1 Introduction

From a practical engineering point of view one of the major design difficulties in
dealing with multiphase flow is that the mass, momentum, and energy transfer rates
and processes can be quite sensitive to the geometric distribution or topology of the
components within the flow. For example, the geometry may strongly effect the inter-
facial area available for mass, momentum, or energy exchange between the phases.
Moreover, the flow within each phase or component will clearly depend on that geo-
metric distribution. Thus we recognize that there is a complicated two-way coupling
between the flow in each of the phases or components and the geometry of the flow (as
well as the rates of change of that geometry). The complexity of this two-way coupling
presents a major challenge in the study of multiphase flows and there is much that
remains to be done before even a superficial understanding is achieved.
An appropriate starting point is a phenomenological description of the geometric
distributions or flow patterns that are observed in common multiphase flows. This
chapter describes the flow patterns observed in horizontal and vertical pipes and
identifies a number of the instabilities that lead to transition from one flow pattern to
another.

7.2 Topologies of Multiphase Flow

7.2.1 Multiphase Flow Patterns

A particular type of geometric distribution of the components is called a flow pattern
or flow regime and many of the names given to these flow patterns (such as annular
flow or bubbly flow) are now quite standard. Usually the flow patterns are recognized
by visual inspection, although other means such as analysis of the spectral content of
the unsteady pressures or the fluctuations in the volume fraction have been devised
for those circumstances in which visual information is difficult to obtain (Jones and
Zuber 1974).
127

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 128 Flow Patterns

For some of the simpler flows, such as those in vertical or horizontal pipes, a
substantial number of investigations have been conducted to determine the dependence
of the flow pattern on component volume fluxes, ( jA , jB ), on volume fraction and on
the fluid properties such as density, viscosity, and surface tension. The results are often
displayed in the form of a flow regime map that identifies the flow patterns occurring
in various parts of a parameter space defined by the component flow rates. The flow
rates used may be the volume fluxes, mass fluxes, momentum fluxes, or other similar
quantities depending on the author. Perhaps the most widely used of these flow pattern
maps is that for horizontal gas/liquid flow constructed by Baker (1954). Summaries
of these flow pattern studies and the various empirical laws extracted from them are
a common feature in reviews of multiphase flow (see, for example, Wallis 1969 or
Weisman 1983).
The boundaries between the various flow patterns in a flow pattern map occur
because a regime becomes unstable as the boundary is approached and growth of
this instability causes transition to another flow pattern. Like the laminar-to-turbulent
transition in single phase flow, these multiphase transitions can be rather unpredictable
because they may depend on otherwise minor features of the flow, such as the roughness
of the walls or the entrance conditions. Hence, the flow pattern boundaries are not
distinctive lines but more poorly defined transition zones.
But there are other serious difficulties with most of the existing literature on flow
pattern maps. One of the basic fluid mechanical problems is that these maps are often
dimensional and therefore apply only to the specific pipe sizes and fluids employed by
the investigator. A number of investigators (for example Baker 1954, Schicht 1969, or
Weisman and Kang 1981) have attempted to find generalized coordinates that would
allow the map to cover different fluids and pipes of different sizes. However, such gen-
eralizations can only have limited value because several transitions are represented in
most flow pattern maps and the corresponding instabilities are governed by different
sets of fluid properties. For example, one transition might occur at a critical Weber
number, whereas another boundary may be characterized by a particular Reynolds
number. Hence, even for the simplest duct geometries, there exist no universal, di-
mensionless flow pattern maps that incorporate the full, parametric dependence of the
boundaries on the fluid characteristics.
Beyond these difficulties there are a number of other troublesome questions. In
single-phase flow it is well established that an entrance length of 30 to 50 diameters is
necessary to establish fully developed turbulent pipe flow. The corresponding entrance
lengths for multiphase flow patterns are less well established and it is quite possible that
some of the reported experimental observations are for temporary or developing flow
patterns. Moreover, the implicit assumption is often made that there exists a unique
flow pattern for given fluids with given flow rates. It is by no means certain that this is
the case. Indeed, in Chapter 16, it is shown that even very simple models of multiphase
flow can lead to conjugate states. Consequently, there may be several possible flow
patterns whose occurence may depend on the initial conditions, specifically on the
manner in which the multiphase flow is generated.

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 7.2 Topologies of Multiphase Flow 129

Figure 7.1. Flow regime map for the horizon-

tal flow of an air/water mixture in a 5.1-cm-
diameter pipe with flow regimes as defined
in Figure 7.2. Hatched regions are observed
regime boundaries, lines are theoretical predic-
tions. Adapted from Weisman (1983).

In summary, there remain many challenges associated with a fundamental under-

standing of flow patterns in multiphase flow and considerable research is necessary
before reliable design tools become available. In this chapter we concentrate on some of
the qualitative features of the boundaries between flow patterns and on the underlying
instabilities that give rise to those transitions.

7.2.2 Examples of Flow Regime Maps

Despite the issues and reservations discussed in the preceding section it is useful
to provide some examples of flow regime maps along with the definitions that help
distinguish the various regimes. We choose to select the first examples from the flows
of mixtures of gas and liquid in horizontal and vertical tubes, mostly because these
flows are of considerable industrial interest. However, many other types of flow regime
maps could be used as examples and some appear elsewhere in this book; examples are
the flow regimes described in the next section and those for granular flows indicated
in Figure 13.5.
We begin with gas/liquid flows in horizontal pipes (see, for example, Hubbard
and Dukler 1966, Wallis 1969, Weisman 1983). Figure 7.1 shows the occurence of
different flow regimes for the flow of an air/water mixture in a horizontal, 5.1-cm
diameter pipe where the regimes are distinguished visually using the definitions in
Figure 7.2. The experimentally observed transition regions are shown by the hatched
areas in Figure 7.1. The solid lines represent theoretical predictions, some of which
are discussed later in this chapter. Note that in a mass flux map like this the ratio

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 130 Flow Patterns

Figure 7.2. Sketches of flow regimes for flow of air/water mixtures in a horizontal, 5.1-cm-diameter pipe.
Adapted from Weisman(1983).

of the ordinate to the abscissa is X /(1 − X ) and therefore the mass quality, X , is
known at every point in the map.
Other examples of flow regime maps for horizontal air/water flow (by different
investigators) are shown in Figures 7.3 and 7.4. These maps plot the volumetric fluxes
rather than the mass fluxes but because the densities of the liquid and gas in these exper-
iments are relatively constant, there is a rough equivalence. Note that in a volumetric

Figure 7.3. A flow regime map for the

flow of an air/water mixture in a hori-
zontal, 2.5-cm-diameter pipe at 25◦ C
and 1 bar. Solid lines and points are
experimental observations of the transi-
tion conditions while the hatched zones
represent theoretical predictions. From
Mandhane et al. (1974).

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 7.2 Topologies of Multiphase Flow 131

Figure 7.4. Same as Figure 7.3 but

showing changes in the flow regime
boundaries for various pipe diameters:
1.25 cm (dotted lines), 2.5 cm (solid
lines), 5 cm (dash-dot lines), and 30 cm
(dashed lines). From Mandhane et al.
(1974).

flux map the ratio of the ordinate to the abscissa is β/(1 − β) and therefore the vol-
umetric quality, β, is known at every point in the map. There are many industrial
processes in which the mass quality is a key flow parameter and therefore mass flux
maps are often preferred.
Figure 7.4 shows how the boundaries were observed to change with pipe diameter.
Moreover, Figures 7.1 and 7.4 appear to correspond fairly closely. Note that both show
well-mixed regimes occuring above some critical liquid flux and above some critical
gas flux; we expand further on this in Section 7.3.1.

7.2.3 Slurry Flow Regimes

As a further example, consider the flow regimes manifest by slurry (solid/liquid mix-
ture) flow in a horizontal pipeline. When the particles are small so that their settling
velocity is much less than the turbulent mixing velocities in the fluid and when the vol-
ume fraction of solids is low or moderate, the flow will be well mixed. This is termed
the homogeneous flow regime (Figure 7.5) and typically occurs only in practical slurry

Figure 7.5. Flow regimes for slurry flow in a horizontal pipeline.

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 132 Flow Patterns

Figure 7.6. A flow regime map for the

flow of an air/water mixture in a vertical,
2.5-cm-diameter pipe showing the ex-
perimentally observed transition regions
hatched; the flow regimes are sketched
in Figure 7.7. Adapted from Weisman
(1983).

pipelines when all the particle sizes are of the order of tens of microns or less. When
somewhat larger particles are present, vertical gradients will occur in the concentra-
tion and the regime is termed heterogeneous; moreover the larger particles will tend
to sediment faster and so a vertical size gradient will also occur. The limit of this
heterogeneous flow regime occurs when the particles form a packed bed in the bottom
of the pipe. When a packed bed develops, the flow regime is known as a saltation flow.
In a saltation flow, solid material may be transported in two ways, either because the
bed moves en masse or because material in suspension above the bed is carried along
by the suspending fluid. Further analyses of these flow regimes, their transitions, and
their pressure gradients are included in Sections 8.2.1, 8.2.2, and 8.2.3. For further
detail, the reader is referred to Shook and Roco (1991), Zandi and Govatos (1967),
and Zandi (1971).

7.2.4 Vertical Pipe Flow

When the pipe is oriented vertically, the regimes of gas/liquid flow are a little different
as illustrated in Figures 7.6 and 7.7 (see, for example, Hewitt and Hall Taylor 1970,
Butterworth and Hewitt 1977, Hewitt 1982, Whalley 1987). Another vertical flow
regime map is shown in Figure 7.8, this one using momentum flux axes rather than
volumetric or mass fluxes. Note the wide range of flow rates in Hewitt and Roberts
(1969) flow regime map and the fact that they correlated both air/water data at atmo-
spheric pressure and steam/water flow at high pressure.
Typical photographs of vertical gas/liquid flow regimes are shown in Figure 7.9.
At low gas volume fractions of the order of a few percentage, the flow is an

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 7.2 Topologies of Multiphase Flow 133

Figure 7.7. Sketches of flow regimes for two-phase flow in a vertical pipe. Adapted from Weisman (1983).

amalgam of individual ascending bubbles (left photograph). Note that the visual ap-
pearance is deceptive; most people would judge the volume fraction to be signifi-
cantly larger than 1%. As the volume fraction is increased (the middle photograph
has α = 4.5%), the flow becomes unstable at some critical volume fraction, which
in the case illustrated is ∼15%. This instability produces large-scale mixing mo-
tions that dominate the flow and have a scale comparable to the pipe diameter. At
still larger volume fractions, large unsteady gas volumes accumulate within these
mixing motions and produce the flow regime known as churn-turbulent flow (right
photograph).
It should be added that flow regime information such as that presented in Fig. 7.8
appears to be valid both for flows that are not evolving with axial distance along the
pipe and for flows, such as those in boiler tubes, in which the volume fraction is

Figure 7.8. The vertical flow regime

map of Hewitt and Roberts (1969) for
flow in a 3.2-cm-diameter tube, validated
for both air/water flow at atmospheric
pressure and steam/water flow at high
pressure.

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 134 Flow Patterns

Figure 7.9. Photographs of air/water flow in a 10.2-cm-diameter vertical pipe (Kytömaa 1987).
(Left) 1% air; (middle) 4.5% air; (right) >15% air.

increasing with axial position. Figure 7.10 provides a sketch of the kind of evo-
lution one might expect in a vertical boiler tube based on the flow regime maps
given above. It is interesting to compare and contrast this flow pattern evolu-
tion with the inverted case of convective boiling surrounding a heated rod in
Figure 6.4.

7.2.5 Flow Pattern Classifications

One of the most fundamental characteristics of a multiphase flow pattern is the extent
to which it involves global separation of the phases or components. At the two ends
of the spectrum of separation characteristics are those flow patterns that are termed
disperse and those that are termed separated. A disperse flow pattern is one in which
one phase or component is widely distributed as drops, bubbles, or particles in the
other continuous phase. Conversely, a separated flow consists of separate, parallel
streams of the two (or more) phases. Even within each of these limiting states there
are various degrees of component separation. The asymptotic limit of a disperse flow
in which the disperse phase is distributed as an infinite number of infinitesimally small
particles, bubbles, or drops is termed a homogeneous multiphase flow. As discussed
in Sections 2.4.2 and 9.2 this limit implies zero relative motion between the phases.
However, there are many practical disperse flows, such as bubbly or mist flow in a
pipe, in which the flow is quite disperse in that the particle size is much smaller
than the pipe dimensions but in which the relative motion between the phases is
significant.
Within separated flows there are similar gradations or degrees of phase separation.
The low-velocity flow of gas and liquid in a pipe that consists of two single-phase

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 7.2 Topologies of Multiphase Flow 135

Figure 7.10. The evolution of the steam/water flow in a vertical boiler tube.

streams can be designated a fully separated flow. Conversely, most annular flows in
a vertical pipe consist of a film of liquid on the walls and a central core of gas that
contains a significant number of liquid droplets. These droplets are an important feature
of annular flow and therefore the flow can only be regarded as partially separated.
To summarize: one of the basic characteristics of a flow pattern is the degree of
separation of the phases into streamtubes of different concentrations. The degree of
separation will, in turn, be determined by (a) some balance between the fluid me-
chanical processes enhancing dispersion and those causing segregation, (b) the initial
conditions or mechanism of generation of the multiphase flow, or (c) some mix of both
effects. In Section 7.3.1 we discuss the fluid mechanical processes referred to in (a).

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 136 Flow Patterns

A second basic characteristic that is useful in classifying flow patterns is the level
of intermittency in the volume fraction. Examples of intermittent flow patterns are
slug flows in both vertical and horizontal pipe flows and the occurrence of interfacial
waves in horizontal separated flow. The first separation characteristic was the degree of
separation of the phases between streamtubes; this second, intermittency, characteristic
can be viewed as the degree of periodic separation in the streamwise direction. The
slugs or waves are kinematic or concentration waves (sometimes called continuity
waves) and a general discussion of the structure and characteristics of such waves is
contained in Chapter 16. Intermittency is the result of an instability in which kinematic
waves grow in an otherwise nominally steady flow to create significant streamwise
separation of the phases.
In the rest of this chapter we describe how these ideas of cross-streamline sep-
aration and intermittency can lead to an understanding of the limits of specific
multiphase flow regimes. The mechanics of limits on disperse flow regimes are
discussed first in Sections 7.3 and 7.4. Limits on separated flow regimes are outlined in
Section 7.5.

7.3 Limits of Disperse Flow Regimes

7.3.1 Disperse Phase Separation and Dispersion

To determine the limits of a disperse phase flow regime, it is necessary to identify
the dominant processes enhancing separation and those causing dispersion. By far
the most common process causing phase separation is due to the difference in the
densities of the phases and the mechanisms are therefore functions of the ratio of
the density of the disperse phase to that of the continuous phase, ρD /ρC . Then the
buoyancy forces caused either by gravity or, in a nonuniform or turbulent flow, by
the Lagrangian fluid accelerations will create a relative velocity between the phases
whose magnitude is denoted by Wp . Using the analysis of Section 2.4.2, we can
conclude that the ratio Wp /U (where U is a typical velocity of the mean flow) is a
function only of the Reynolds number, Re = 2U R/νC , and the parameters X and Y
are defined by Eqs. (2.91) and (2.92). The particle size, R, and the streamwise extent of
the flow, , both occur in the dimensionless parameters Re, X , and Y . For low velocity
flows in which U 2 /  g,  is replaced by g/U 2 and hence a Froude number, g R/U 2 ,
rather than R/ appears in the parameter X . This then establishes a velocity, Wp , that
characterizes the relative motion and therefore the phase separation due to density
differences.
As an aside we note that there are some fluid mechanical phenomena that can
cause phase separation even in the absence of a density difference. For example, Ho
and Leal (1974) explored the migration of neutrally buoyant particles in shear flows
at low Reynolds numbers. These effects are usually suffciently weak compared with
those due to density differences that they can be neglected in many applications.

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 7.3 Limits of Disperse Flow Regimes 137

Figure 7.11. Bubbly flow around a NACA 4412 hydrofoil (10-cm chord) at an angle of attack; flow is
from left to right. From the work of Ohashi et al. (1990), reproduced with the author’s permission.

In a quiescent multiphase mixture the primary mechanism of phase separation is

sedimentation (see Chapter 16) though more localized separation can also occur as a
result of the inhomogeneity instability described in Section 7.4. In flowing mixtures the
mechanisms are more complex and, in most applications, are controlled by a balance
between the buoyancy/gravity forces and the hydrodynamic forces. In high-Reynolds-
number, turbulent flows, the turbulence can cause either dispersion or segregation.
Segregation can occur when the relaxation time for the particle or bubble is compa-
rable with the typical time of the turbulent fluid motions. When ρD /ρC  1 as, for
example, with solid particles suspended in a gas, the particles are centrifuged out of
the more intense turbulent eddies and collect in the shear zones in between (see, for
example, Squires and Eaton 1990, Elghobashi and Truesdell 1993). Conversely, when
ρD /ρC  1 as, for example, with bubbles in a liquid, the bubbles tend to collect in
regions of low pressure such as in the wake of a body or in the centers of vortices
(see, for example, Pan and Banerjee 1997). We previously included a photograph
(Figure 1.6) showing heavier particles centrifuged out of vortices in a turbulent chan-
nel flow. Here, as a counterpoint, we include the photograph, Figure 7.11, from Ohashi
et al. (1990) showing the flow of a bubbly mixture around a hydrofoil. Note the region
of higher void fraction (more than four times the upstream void fraction according to
the measurements) in the wake on the suction side of the foil. This accumulation of
bubbles on the suction side of a foil or pump blade has importance consequences for
performance as discussed in Section 7.3.3.
Counteracting the above separation processes are dispersion processes. In many
engineering contexts the principal dispersion is caused by the turbulent or other un-
steady motions in the continuous phase. Figure 7.11 also illustrates this process for the
concentrated regions of high void fraction in the wake are dispersed as they are carried

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 138 Flow Patterns

downstream. The shear created by unsteady velocities can also cause either fission or
fusion of the disperse phase bubbles, drops, or particles, but we delay discussion of
this additional complexity until the next section. For the present it is necessary only
to characterize the mixing motions in the continuous phase by a typical velocity, Wt .
Then the degree of separation of the phases will clearly be influenced by the relative
magnitudes of Wp and Wt or, specifically, by the ratio Wp /Wt . Disperse flow will
occur when Wp /Wt  1 and separated flow when Wp /Wt  1. The corresponding
flow pattern boundary should be given by some value of Wp /Wt of order unity. For
example, in slurry flows in a horizontal pipeline, Thomas (1962) suggested a value of
Wp /Wt of 0.2 based on his data.

7.3.2 Example: Horizontal Pipe Flow

As a quantitative example, we pursue the case of the flow of a two-component mixture
in a long horizontal pipe. The separation velocity, Wp , due to gravity, g, would then
be given qualitatively by Eq. (2.74) or (2.83), namely


2R 2 g ρ
Wp = if 2Wp R/νC  1 (7.1)
9νC ρC
or
 1
2 Rg ρ 2
Wp = if 2Wp R/νC  1, (7.2)
3 CD ρC
where R is the particle, droplet, or bubble radius; νC and ρC are the kinematic viscosity
and density of the continuous fluid; and ρ is the density difference between the
components. Furthermore, the typical turbulent velocity will be some function of the
1
friction velocity, (τw /ρC ) 2 , and the volume fraction, α, of the disperse phase. The effect
of α is less readily quantified so, for the present, we concentrate on dilute systems
(α  1) in which

 12 
 12
τw d dp
Wt ≈ = − , (7.3)
ρC 4ρC ds
where d is the pipe diameter and dp/ds is the pressure gradient. Then the transition
condition, Wp /Wt = K (where K is some number of order unity), can be rewritten as
follows:


dp 4ρC
− ≈ 2 Wp2 (7.4)
ds K d


16 ρC R 4 g 2 ρ 2
≈ for 2Wp R/νC  1 (7.5)
81K 2 νC2 d ρC


32 ρC Rg ρ
≈ for 2Wp R/νC  1. (7.6)
3K 2 CD d ρC

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 7.3 Limits of Disperse Flow Regimes 139

In summary, the expression on the right-hand side of Eq. (7.5) [or (7.6)] yields
the pressure drop at which Wp /Wt exceeds the critical value of K and the parti-
cles will be maintained in suspension by the turbulence. At lower values of the
pressure drop the particles will settle out and the flow will become separated and
stratified.
This criterion on the pressure gradient may be converted to a criterion on the flow
rate by using some version of the turbulent pipe flow relation between the pressure
gradient and the volume flow rate, j. For example, one could conceive of using, as
a first approximation, a typical value of the turbulent friction factor, f = τw / 12 ρC j 2
(where j is the total volumetric flux). In the case of 2Wp R/νC  1, this leads to a
critical volume flow rate, j = jc , given by the following:
 1
8 g D ρ 2
jc = . (7.7)
3K 2 f CD ρC
With 8/3K 2 f replaced by an empirical constant, this is the general form of the crit-
ical flow rate suggested by Newitt et al. (1955) for horizontal slurry pipeline flow;
for j > jc the flow regime changes from saltation flow to heterogeneous flow (see
Figure 7.5). Alternatively, one could write this nondimensionally using a Froude num-
1
ber defined as Fr = jc /(gd) 2 . Then the criterion yields a critical Froude number given
by the following:

8 ρ
Fr2 = (7.8)
3K 2f C D ρC
1
If the common expression for the turbulent friction factor, namely f = 0.31/( jd/νC ) 4
is used in Eq. (7.7) that expression becomes the following:
 4
 17.2 g Rd 14 ρ  7
jc = . (7.9)
 K 2 CD ν 14 ρC 
C

A numerical example will help relate this criterion to the boundary of the disperse
phase regime in the flow regime maps. For the case of Figure 7.3 and using for
simplicity, K = 1 and CD = 1, then with a drop or bubble size, R = 3 mm, Eq. (7.9)
gives a value of jc of 3 m/s when the continuous phase is liquid (bubbly flow) and a
value of 40 m/s when the continuous phase is air (mist flow). These values are in good
agreement with the total volumetric flux at the boundary of the disperse flow regime
in Figure 7.3, which, at low jG , is about 3 m/s and at higher jG (volumetric qualities
above 0.5) is about 30–40 m/s.
Another approach to the issue of the critical velocity in slurry pipeline flow is
to consider the velocity required to fluidize a packed bed in the bottom of the pipe
(see, for example, Durand and Condolios 1952 or Zandi and Govatos 1967). This is
described further in Section 8.2.3.

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 140 Flow Patterns

7.3.3 Particle Size and Particle Fission

In the preceding sections, the transition criteria determining the limits of the disperse
flow regime included the particle, bubble, or drop size or, more specifically, the dimen-
sionless parameter 2R/d as illustrated by the criteria of Eqs. (7.5), (7.6), and (7.9).
However, these criteria require knowledge of the size of the particles, 2R, and this
is not always accessible particularly in bubbly flow. Even when there may be some
knowledge of the particle or bubble size in one region or at one time, the various pro-
cesses of fission and fusion need to be considered in determining the appropriate 2R
for use in these criteria. One of the serious complications is that the size of the particles,
bubbles, or drops is often determined by the flow itself because the flow shear tends
to cause fission and therefore limit the maximum size of the surviving particles. Then
the flow regime may depend on the particle size that in turn depends on the flow and
this two-way interaction can be difficult to unravel. Figure 7.11 illustrates this problem
because one can observe many smaller bubbles in the flow near the suction surface and
in the wake that clearly result from fission in the highly sheared flow near the suction
surface. Another example from the flow in pumps is described in the next section.
When the particles are very small, a variety of forces may play a role in determining
the effective particle size and some comments on these are included later in Section
7.3.7. But often the bubbles or drops are sufficiently large that the dominant force
resisting fission is due to surface tension while the dominant force promoting fission
is the shear in the flow. We confine the present discussion to these circumstances.
Typical regions of high shear occur in boundary layers, in vortices or in turbulence.
Frequently, the larger drops or bubbles are fissioned when they encounter regions
of high shear and do not subsequently coalesce to any significant degree. Then, the
characteristic force resisting fission would be given by S R, whereas the typical shear
force causing fission might be estimated in several ways. For example, in the case of
pipe flow the typical shear force could be characterized by τw R 2 . Then, assuming that
the flow is initiated with larger particles that are then fissioned by the flow, we would
estimate that R = S/τw . This is used in the next section to estimate the limits of the
bubbly or mist flow regime in pipe flows.
In other circumstances, the shearing force in the flow might be described by
ρC (γ̇ R)2 R 2 where γ̇ is the typical shear rate and ρC is the density of the continu-
ous phase. This expression for the fission force assumes a high Reynolds number in
the flow around the particle or explicitly that ρC γ̇ R 2 /µC  1 where µC is the dynamic
viscosity of the continuous phase. But if ρC γ̇ R 2 /µC  1, then a more appropriate
estimate of the fission force would be µC γ̇ R 2 . Consequently, the maximum particle
size, Rm , one would expect to see in the flow in these two regimes would be as follows:

 
S
Rm = for ρC γ̇ R 2 /µC  1
µC γ̇
  13
S
or for ρC γ̇ R 2 /µC  1, (7.10)
ρC γ̇ 2

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 7.3 Limits of Disperse Flow Regimes 141

Figure 7.12. A bubbly air/water mixture (volume fraction about 4%) entering an axial flow impeller (a
10.2-cm-diameter scale model of the SSME low-pressure liquid oxygen impeller) from the right. The
inlet plane is roughly in the center of the photograph and the tips of the blades can be seen to the left of
the inlet plane.

respectively. Note that in both instances the maximum size decreases with increasing
shear rate.

7.3.4 Examples of Flow-Determined Bubble Size

An example of the use of the above relations can be found in the important area of
two-phase pump flows and we quote here data from studies of the pumping of bubbly
liquids. The issue here is the determination of the volume fraction at which the pump
performance is seriously degraded by the presence of the bubbles. It transpires that,
in most practical pumping situations, the turbulence and shear at inlet and around the
leading edges of the blades of the pump (or other turbomachine) tend to fission the
bubbles and thus determine the size of the bubbles in the blade passages. An illustration
is included in Figure 7.12 which shows an air/water mixture progressing through an
axial flow impeller; the bubble size downstream of the inlet plane is much smaller that
that approaching the impeller.
The size of the bubbles within the blade passages is important because it is the
migration and coalescence of these bubbles that appear to cause degradation in the
performance. Because the velocity of the relative motion depends on the bubble size,
it follows that the larger the bubbles the more likely it is that large voids will form
within the blade passage due to migration of the bubbles toward regions of lower
pressure (Furuya 1985, Furuya and Maekawa 1985). As Patel and Runstadler (1978)
observed during experiments on centrifugal pumps and rotating passages, regions
of low pressure occur not only on the suction sides of the blades but also under
the shroud of a centrifugal pump. These large voids or gas-filled wakes can cause

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 142 Flow Patterns

Figure 7.13. The bubble sizes, Rm , observed in the blade passages of centrifugal and axial flow pumps
as a function of Weber number, where h is the blade spacing (adapted from Murakami and Minemura
1978).

substantial changes in the deviation angle of the flow leaving the impeller and hence
lead to substantial degradation in the pump performance.
The key is therefore the size of the bubbles in the blade passages and some valuable
data on this has been compiled by Murakami and Minemura (1977, 1978) for both
axial and centrifugal pumps. This is summarized in Figure 7.4, where the ratio of
the observed bubble size, Rm , to the blade spacing, h, is plotted against the Weber
number, We = ρCU 2 h/S (U is the blade tip velocity). Rearranging the first version
of Eq. (7.10), estimating that the inlet shear is proportional to U/ h and adding a
proportionality constant, C, because the analysis is qualitative, we would expect that
1
Rm = C/We 3 . The dashed lines in Figure 7.13 are examples of this prediction and
exhibit behavior very similar to the experimental data. In the case of the axial pumps,
the effective value of the coefficient, C = 0.15.
A different example is provided by cavitating flows in which the highest shear rates
occur during the collapse of the cavitation bubbles. As discussed in Section 5.2.3,
these high shear rates cause individual cavitation bubbles to fission into many smaller
fragments so that the bubble size emerging from the region of cavitation bubble col-
lapse is much smaller than the size of the bubbles entering that region. The phenomenon
is exemplified by Figure 7.14, which shows the growth of the cavitating bubbles on
the suction surface of the foil, the collapse region near the trailing edge and the much
smaller bubbles emerging from the collapse region. Some analysis of the fission due
to cavitation bubble collapse is contained in Brennen (2002).

7.3.5 Bubbly or Mist Flow Limits

Returning now to the issue of determining the boundaries of the bubbly (or mist
flow) regime in pipe flows, and using the expression R = S/τw for the bubble size

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 7.3 Limits of Disperse Flow Regimes 143

Figure 7.14. Traveling bubble cavitation on the surface of a NACA 4412 hydrofoil at zero incidence
angle, a speed of 13.7 m/s, and a cavitation number of 0.3. The flow is from left to right, the leading edge
of the foil is just to the left of the white glare patch on the surface, and the chord is 7.6 cm (Kermeen
1956).

in Eq. (7.6), the transition between bubbly disperse flow and separated (or partially
separated flow) is described by the following relation:
1
 − 14   14
− ddsp 2
S 64
= = constant. (7.11)
gρ gd ρ
2 3K 2 CD

This is the analytical form of the flow regime boundary suggested by Taitel and Dukler
(1976) for the transition from disperse bubbly flow to a more separated state. Taitel and
Dukler also demonstrate that when the constant in Eq. (7.11) is of the order of unity, the
boundary agrees well with that observed experimentally by Mandhane et al. (1974).
This agreement is shown in Figure 7.3. The same figure serves to remind us that there
are other transitions that Taitel and Dukler were also able to model with qualitative
arguments. They also demonstrate, as mentioned earlier, that each of these transitions
typically scale differently with the various nondimensional parameters governing the
characteristics of the flow and the fluids.

7.3.6 Other Bubbly Flow Limits

As the volume fraction of gas or vapor is increased, a bubbly flow usually transitions
to a mist flow, a metamorphosis that involves a switch in the continuous and disperse
phases. However, there are several additional comments on this metamorphosis that
need to be noted.
First, at very low flow rates, there are circumstances in which this transition does
not occur at all and the bubbly flow becomes a foam. Though the precise conditions
necessary for this development are not clear, foams and their rheology have been the

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 144 Flow Patterns

subject of considerable study. The mechanics of foams are beyond the scope of this
book; the reader is referred to the review of Kraynik (1988) and the book of Weaire
and Hutzler (2001).
Second, though it is rarely mentioned, the reverse transition from mist flow to bub-
bly flow as the volume fraction decreases involves energy dissipation and an increase
in pressure. This transition has been called a mixing shock (Witte 1969) and typically
occurs when a droplet flow with significant relative motion transitions to a bubbly flow
with negligible relative motion. Witte (1969) has analyzed these mixing shocks and
obtains expressions for the compression ratio across the mixing shock as a function
of the upstream slip and Euler number.

7.3.7 Other Particle Size Effects

In Sections 7.3.3 and 7.3.5 we outlined one class of circumstances in which bubble
fission is an important facet of the disperse phase dynamics. It is, however, important
to add, even if briefly, that there are many other mechanisms for particle fission and fu-
sion that may be important in a disperse phase flow. When the particles are submicron
or micron sized, intermolecular and electromagnetic forces can become critically im-
portant in determining particle aggregation in the flow. These phenomena are beyond
the scope of this book and the reader is referred to texts such as Friedlander (1977) or
Flagan and Seinfeld (1988) for information on the effects these forces have on flows
involving particles and drops. It is, however, valuable to add that gas/solid suspension
flows with larger particles can also exhibit important effects as a result of electrical
charge separation and the forces that those charges create between particles or between
the particles and the walls of the flow. The process of electrification or charge separa-
tion is often a very important feature of such flows (Boothroyd 1971). Pneumatically
driven flows in grain elevators or other devices can generate huge electropotential
differences (as large as hundreds of kilovolts) that can, in turn, cause spark discharges
and consequently dust explosions. In other devices, particularly electrophotographic
copiers, the charge separation generated in a flowing toner/carrier mixture is a key
feature of such devices. Electromagnetic and intermolecular forces can also play a role
in determining the bubble or droplet size in gas/liquid flows (or flows of immiscible
liquid mixtures).

7.4 Inhomogeneity Instability

In Section 7.3.1 we presented a qualitative evaluation of phase separation processes
driven by the combination of a density difference and a fluid acceleration. Such a
combination does not necessarily imply separation within a homogeneous quiescent
mixture (except through sedimentation). However, it transpires that local phase sepa-
ration may also occur through the development of an inhomogeneity instability whose
origin and consequences we describe in the next two sections.

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 7.4 Inhomogeneity Instability 145

7.4.1 Stability of Disperse Mixtures

It transpires that a homogeneous, quiescent multiphase mixture may be internally un-
stable as a result of gravitationally induced relative motion. This instability was first
described for fluidized beds by Jackson (1963). It results in horizontally oriented, ver-
tically propagating volume fraction waves or layers of the disperse phase. To evaluate
the stability of a uniformly dispersed two component mixture with uniform relative
velocity induced by gravity and a density difference, Jackson constructed a model
consisting of the following system of equations:
1. The number continuity equation [Eq. (1.30)] for the particles (density, ρD , and
volume fraction, αD = α):
∂α ∂(αu D )
+ = 0, (7.12)
∂t ∂y
where all velocities are in the vertically upward direction.
2. Volume continuity for the suspending fluid (assuming constant density, ρC , and
zero mass interaction, IN = 0):
∂α ∂((1 − α)u C )
− = 0. (7.13)
∂t ∂y
3. Individual phase momentum equations [Eq. (1.42)] for both the particles and the
fluid assuming constant densities and no deviatoric stress:
 
∂u D ∂u D
ρD α + uD = −αρD g + FD (7.14)
∂t ∂y
 
∂u C ∂u C ∂p
ρC (1 − α) + uC = −(1 − α)ρC g − − FD . (7.15)
∂t ∂y ∂y
4. A force interaction term of the form given by Eq. (1.44). Jackson constructs a

component, FDk , due to the relative motion of the form
FD = q(α)(1 − α)(u C − u D ), (7.16)
where q is assumed to be some function of α. Note that this is consistent with a
low Reynolds number flow.
Jackson then considered solutions of these equations that involve small, linear
perturbations or waves in an otherwise homogeneous mixture. Thus the flow was
decomposed into the following:
1. A uniform, homogeneous fluidized bed in which the mean values of u D and u C
are respectively zero and some adjustable constant. To maintain generality, we will
characterize the relative motion by the drift flux, jCD = α(1 − α)u C .
2. An unsteady linear perturbation in the velocities, pressure, and volume fraction
of the form exp{iκ y + (ζ − iω)t} that models waves of wavenumber, κ, and

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 146 Flow Patterns

frequency, ω, traveling in the y direction with velocity ω/κ and increasing in

amplitude at a rate given by ζ.
Substituting this decomposition into the system of equations described above yields
the following expression for (ζ − iω):
jCD 1
(ζ − iω) = ±K 2 1 + 4i K 3 + 4K 1 K 32 − 4i K 3 (1 + K 1 )K 4 2
g
− K 2 (1 + 2i K 3 ) (7.17)
where the constants K 1 through K 3 are given by the following:
ρD (1 − α) (ρD − ρC )α(1 − α)
K1 = ; K2 =
ρC α 2 {ρD (1 − α) + ρC α}
κ jCD
2
K3 = (7.18)
gα(1 − α)2 {ρD /ρC − 1}
and K 4 is given by the following:
α(1 − α) dq
K 4 = 2α − 1 + . (7.19)
q dα
It transpires that K 4 is a critical parameter in determining the stability and it, in turn,
depends on how q, the factor of proportionality in Eq. (7.16), varies with α. Here
we examine two possible functions, q(α). The Carman–Kozeny Eq. (2.96) for the
pressure drop through a packed bed is appropriate for slow viscous flow and leads
to q ∝ α 2 /(1 − α)2 ; from Eq. (7.19) this yields K 4 = 2α + 1 and is an example of
low-Reynolds-number flow. As a representative example of higher Reynolds number
flow we take the relation 2.100 due to Wallis (1969) and this leads to q ∝ α/(1 −
α)b−1 (recall Wallis suggests b = 3); this yields K 4 = bα. We examine both of these
examples of the form of q(α).
Note that the solution [Eq. (7.17)] yields the nondimensional frequency and growth
rate of waves with wavenumber, κ, as functions of just three dimensionless variables,
the volume fraction, α, the density ratio, ρD /ρC , and the relative motion parameter,
1
jCD /(g/κ) 2 , similar to a Froude number. Note also that Eq. (7.17) yields two roots for
the dimensionless frequency, ωjCD /g, and growth rate, ζ jCD /g. Jackson demonstrates
that the negative sign choice is an attenuated wave; consequently we focus exclusively
on the positive sign choice that represents a wave that propagates in the direction of the
drift flux, jCD , and grows exponentially with time. It is also easy to see that the growth
rate tends to infinity as κ → ∞. However, it is meaningless to consider wavelengths
less than the interparticle distance and therefore the focus should be on waves of this
order because they will predominate. Therefore, in the discussion below, it is assumed
that the κ −1 values of primary interest are of the order of the typical interparticle
distance.
Figure 7.15 presents typical dimensionless growth rates for various values of
1
the parameters α, ρD /ρC , and jCD /(g/k) 2 for both the Carman–Kozeny and Wallis

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 7.4 Inhomogeneity Instability 147

Figure 7.15. The dimensionless growth rate

ζ jCD /g plotted against the parameter jCD /
1
(g/κ) 2 for various values of α and ρD /ρC and
for both K 4 = 2α + 1 and K 4 = 3α.

expressions for K 4 . In all cases the growth rate increases with the wavenumber κ,
confirming the fact that the fastest growing wavelength is the smallest that is relevant.
We note, however, that a more complete linear analysis by Anderson and Jackson
(1968) (see also Homsy et al. 1980, Jackson 1985, Kytömaa 1987) that includes vis-
cous effects yields a wavelength that has a maximum growth rate. Figure 7.15 also
demonstrates that the effect of void fraction is modest; although the lines for α = 0.5
lie below those for α = 0.1 this must be weighed in conjunction with the fact that
the interparticle distance is greater in the latter case. Gas and liquid fluidized beds
are typified by ρD /ρC values of 3000 and 3 respectively; because the lines for these
two cases are not far apart, the primary difference is the much larger values of jCD
in gas-fluidized beds. Everything else being equal, increasing jCD means following a
line of slope 1 in Figure 7.15 and this implies much larger values of the growth rate
in gas-fluidized beds. This is in accord with the experimental observations.
As a postscript, it must be noted that the above analysis leaves out many effects that
may be consequential. As previously mentioned, the inclusion of viscous effects is
important at least for lower Reynolds number flows. At higher particle Reynolds
numbers, even more complex interactions can occur as particles encounter the
wakes of other particles. For example, Fortes et al. (1987) demonstrated the com-
plexity of particle/particle interactions under those circumstances and Joseph (1993)

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 148 Flow Patterns

provides a summary of how the inhomogeneities or volume fraction waves evolve with
such interactions. General analyses of kinematic waves are contained in Chapter 16
and the reader is referred to that chapter for details.

7.4.2 Inhomogeneity Instability in Vertical Flows

In vertical flows, the inhomogeneity instability described in the previous section will
mean the development of intermittency in the volume fraction. The short-term result
of this instability is the appearance of vertically propagating, horizontally oriented
kinematic waves (see Chapter 16) in otherwise nominally steady flows. They have been
most extensively researched in fluidized beds but have also be observed experimentally
in vertical bubbly flows by Bernier (1982), Boure and Mercadier (1982), Kytomaa and
Brennen (1990) (who also examined solid/liquid mixtures at large Reynolds numbers)
and analyzed by Biesheuvel and Gorissen (1990). (Some further comment on these
bubbly flow measurements is contained in Section 16.2.3.)
As they grow in amplitude these wavelike volume fraction perturbations seem to
evolve in several ways depending on the type of flow and the manner in which it is
initiated. In turbulent gas/liquid flows they result in large gas volumes or slugs with a
size close to the diameter of the pipe. In some solid/liquid flows they produce a series
of periodic vortices, again with a dimension comparable with that of the pipe diameter.
But the long-term consequences of the inhomogeneity instability have been most care-
fully studied in the context of fluidized beds. Following the work of Jackson (1963),
El-Kaissy and Homsy (1976) studied the evolution of the kinematic waves experimen-
tally and observed how they eventually lead, in fluidized beds, to three-dimensional
structures known as bubbles. These are not gas bubbles but three-dimensional, bub-
blelike zones of low particle concentration that propagate upward through the bed
while their structure changes relatively slowly. They are particularly evident in wide
fluidized beds where the lateral dimension is much larger than the typical interparti-
cle distance. Sometimes bubbles are directly produced by the sparger or injector that
creates the multiphase flow. This tends to be the case in gas-fluidized beds where, as
illustrated in the preceding section, the rate of growth of the inhomogeneity is much
greater than in liquid fluidized beds and thus bubbles are instantly formed.
Because of their ubiquity in industrial processes, the details of the three-
dimensional flows associated with fluidized-bed bubbles have been extensively stud-
ied both experimentally (see, for example, Davidson and Harrison 1963, Davidson
et al. 1985) and analytically (Jackson 1963, Homsy et al. 1980). Roughly spherical
or spherical cap in shape, these zones of low solids volume fraction always rise in a
fluidized bed (see Figure 7.16). When the density of bubbles is low, single bubbles
are observed to rise with a velocity, WB , given empirically by Davidson and Harrison
(1963) as
1 1
WB = 0.71g 2 VB6 , (7.20)

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 7.4 Inhomogeneity Instability 149

Figure 7.16. (Left) X-ray image of fluidized bed bubble (about 5 cm in diameter) in a bed of glass beads
(courtesy of P.T. Rowe). (Right) View from above of bubbles breaking the surface of a sand/air fluidized
bed (courtesy of J.F. Davidson).

where VB is the volume of the bubble. Both the shape and rise velocity have many
similarities to the spherical-cap bubbles discussed in Section 3.2.2. The rise veloc-
ity, WB may be either faster or slower than the upward velocity of the suspending
fluid, u C , and this implies two types of bubbles that Catipovic et al. (1978) call
fast and slow bubbles respectively. Figure 7.17 qualitatively depicts the nature of
the streamlines of the flow relative to the bubbles for fast and slow bubbles. The
same article provides a flow regime map, Figure 7.18 indicating the domains of

Figure 7.17. Sketches of the fluid streamlines relative to a fluidized bed bubble of low volume fraction
for a fast bubble (left) and a slow bubble. Adapted from Catipovic et al. (1978).

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 150 Flow Patterns

Figure 7.18. Flow regime map for flu-

idized beds with large particles (diam-
eter, D), where (u C )min is the minimum
fluidization velocity and H is the height
of the bed. Adapted from Catipovic et al.
(1978).

fast bubbles, slow bubbles, and rapidly growing bubbles. When the particles are
smaller other forces become important, particularly those that cause particles to
stick together. In gas-fluidized beds the flow regime map of Geldart (1973), re-
produced as Figure 7.19, is widely used to determine the flow regime. With very
small particles (Group C) the cohesive effects dominate and the bed behaves like
a plug, although the suspending fluid may create holes in the plug. With some-
what larger particles (Group A), the bed exhibits considerable expansion before
bubbling begins. Group B particles exhibit bubbles as soon as fluidization begins
(fast bubbles) and, with even larger particles (Group D), the bubbles become slow
bubbles.
Aspects of the flow regime maps in Figures 7.18 and 7.19 qualitatively reflect the
results of the instability analysis of the last section. Larger particles and larger fluid
velocities imply larger jCD values and therefore, according to instability analysis,
larger growth rates. Thus, in the upper right side of both figures we find rapidly
growing bubbles. Moreover, in the instability analysis it transpires that the ratio of the
wave speed, ω/κ (analogous to the bubble velocity) to the typical fluid velocity, jCD ,
1
is a continuously decreasing function of the parameter, jCD /(g/κ) 2 . Indeed, ω/jCD κ

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 7.5 Limits on Separated Flow 151

Figure 7.19. Flow regime map for fluidized beds with small particles (diameter, D). Adapted from Geldart
(1973).

1
decreases from values greater than unity to values less than unity as jCD /(g/κ) 2
increases. This is entirely consistent with the progression from fast bubbles for small
particles (small jCD ) to slow bubbles for larger particles.
For further details on bubbles in fluidized beds the reader is referred to the exten-
sive literature, including the books of Zenz and Othmer (1960), Cheremisinoff and
Cheremisinoff (1984), Davidson et al. (1985), and Gibilaro (2001).

7.5 Limits on Separated Flow

We now leave disperse flow limits and turn to the mechanisms that limit separated
flow regimes.

7.5.1 Kelvin–Helmoltz Instability

Separated flow regimes such as stratified horizontal flow or vertical annular flow can
become unstable when waves form on the interface between the two fluid streams (sub-
scripts 1 and 2). As indicated in Figure 7.20, the densities of the fluids will be denoted by
ρ1 and ρ2 and the velocities by u 1 and u 2 . If these waves continue to grow in amplitude
they will cause a transition to another flow regime, typically one with greater inter-
mittency and involving plugs or slugs. Therefore, to determine this particular bound-
ary of the separated flow regime, it is necessary to investigate the potential growth
of the interfacial waves, whose wavelength is denoted by λ (wavenumber, κ = 2π/λ).
Studies of such waves have a long history originating with the work of Kelvin and

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 152 Flow Patterns

Figure 7.20. Sketch showing the notation for Kelvin–Helmholtz instability.

Helmholtz and the phenomena they revealed have come to be called Kelvin–Helmholtz
instabilities (see, for example, Yih 1965). In general this class of instabilities involves
the interplay between at least two of the following three types of forces:
r a buoyancy force due to gravity and proportional to the difference in the densities
of the two fluids. This can be characterized by g3 ρ, where ρ = ρ1 − ρ2 , g
is the acceleration due to gravity, and  is a typical dimension of the waves. This
force may be stabilizing or destabilizing depending on the orientation of gravity,
g, relative to the two fluid streams. In a horizontal flow in which the upper fluid
is lighter than the lower fluid the force is stabilizing. When the reverse is true
the buoyancy force is destabilizing and this causes Rayleigh–Taylor instabilities.
When the streams are vertical as in vertical annular flow the role played by the
buoyancy force is less clear.
r a surface tension force characterized by S that is always stabilizing.
r a Bernoulli effect that implies a change in the pressure acting on the interface
caused by a change in velocity resulting from the displacement, a, of that surface.
For example, if the upward displacement of the point A in Figure 7.21 were to
cause an increase in the local velocity of fluid 1 and a decrease in the local velocity
of fluid 2, this would imply an induced pressure difference at the point A that would
increase the amplitude of the distortion, a. Such Bernoulli forces depend on the
difference in the velocity of the two streams, u = u 1 − u 2 , and are characterized
by ρ( u)2 2 , where ρ and  are a characteristic density and dimension of the
flow.

Figure 7.21. Sketch showing the notation for stratified flow instability.

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 7.5 Limits on Separated Flow 153

The interplay between these forces is most readily illustrated by a simple exam-
ple. Neglecting viscous effects, one can readily construct the planar, incompressible
potential flow solution for two semi-infinite horizontal streams separated by a plane
horizontal interface (as in Figure 7.20) on which small-amplitude waves have formed.
Then it is readily shown (Lamb 1879, Yih 1965) that Kelvin–Helmholtz instability
will occur when
gρ ρ1 ρ2 ( u)2
+ Sκ − < 0. (7.21)
κ ρ1 + ρ2
The contributions from the three previously mentioned forces are self-evident. Note
that the surface tension effect is stabilizing because that term is always positive, the
buoyancy effect may be stabilizing or destabilizing depending on the sign of ρ,
and the Bernoulli effect is always destabilizing. Clearly, one subset of this class of
Kelvin–Helmholtz instabilities are the Rayleigh–Taylor instabilities that occur in the
absence of flow ( u = 0) when ρ is negative. In that static case, the above relation
shows that the interface is unstable to all wave numbers less than the critical value,
κ = κc , where

1
g(− ρ) 2
κc = . (7.22)
S
In the next two sections we focus on the instabilities induced by the destabilizing
Bernoulli effect for these can often cause instability of a separated flow regime.

7.5.2 Stratified Flow Instability

As a first example, consider the stability of the horizontal stratified flow depicted
in Figure 7.21, where the destabilizing Bernoulli effect is primarily opposed by a
stabilizing buoyancy force. An approximate instability condition is readily derived by
observing that the formation of a wave (such as that depicted in Figure 7.21) will lead
to a reduced pressure, pA , in the gas in the orifice formed by that wave. The reduction
below the mean gas pressure, p̄ G , is given by Bernoulli’s equation as follows:
pA − p̄ G = −ρG u 2G a/ h, (7.23)
provided a  h. The restraining pressure is given by the buoyancy effect of the el-
evated interface, namely (ρL − ρG )ga. It follows that the flow will become unstable
when
u 2G > gh ρ/ρG . (7.24)
In this case the liquid velocity has been neglected because it is normally small com-
pared with the gas velocity. Consequently, the instability criterion provides an upper
limit on the gas velocity that is, in effect, the velocity difference. Taitel and Dukler
(1976) compared this prediction for the boundary of the stratified flow regime in a

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 154 Flow Patterns

horizontal pipe of diameter, d, with the experimental observations of Mandhane et al.

(1974) and found substantial agreement. This can be demonstrated by observing that,
from Eq. (7.24),
1
jG = αu G = C(α)α(gd ρ/ρG ) 2 , (7.25)
1
where C(α) = (h/d) 2 is some simple monotonically increasing function of α that
depends on the pipe cross section. For example, for the 2.5-cm pipe of Figure 7.3 the
1
factor (gd ρ/ρG ) 2 in Eq. (7.25) will have a value of approximately 15 m/s. As shown
in Figure 7.3, this is in close agreement with the value of jG at which the flow at low
jL departs from the stratified regime and begins to become wavy and then annular.
Moreover, the factor C(α)α should decrease as jL increases and, in Figure 7.3, the
boundary between stratified flow and wavy flow also exhibits this decrease.

7.5.3 Annular Flow Instability

As a second example consider vertical annular flow that becomes unstable when the
Bernoulli force overcomes the stabilizing surface tension force. From Eq. (7.21), this
implies that disturbances with wavelengths greater than a critical value, λc , will be
unstable and that
λc = 2πS(ρ1 + ρ2 )/ρ1 ρ2 ( u)2 . (7.26)
For a liquid stream and a gas stream (as is normally the case in annular flow) and with
ρL  ρG this becomes the following:
λc = 2πS/ρG ( u)2 . (7.27)
Now consider the application of this criterion to the flow regime maps for vertical pipe
flow included in Figures 7.6 and 7.8. We examine the stability of a well-developed
annular flow at a high gas volume fraction where u ≈ jG . Then, for a water/air
mixture, Eq. (7.27) predicts critical wavelengths of 0.4 and 40 cm for jG = 10 m/s
and jG = 1 m/s respectively. In ther words, at low values of jG only larger wavelengths
are unstable and this seems to be in accord with the breakup of the flow into large
slugs. Conversely, at higher jG flow rates, even quite small wavelengths are unstable
and the liquid gets torn apart into the small droplets carried in the core gas flow.

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