1

Stability analysis of inclined stratified two-phase gas-liquid flow

Yacine Salhi a,b, El-Khider Si-Ahmed a,c , Jack Legrand c and Gérard Degrez b

a Laboratoire de Mécanique des Fluides Théorique et Appliquée, Faculté de Physique,

U.S.T.H.B., Alger, Algérie

bService Aéro-Thermo-Mécanique Faculté des Sciences Appliquées Université Libre de Bruxelles

cGEPEA, Université de Nantes, CNRS, UMR6144, CRTT-BP 406, 44602 Saint-Nazaire, France

Abstract

The present investigation involves the modeling of gas-liquid interface in a two-phase stratified

flow through a horizontal or nearly-horizontal circular duct. The most complete and fundamental

model used for these calculations is known as the one dimensional two-fluid model. It is the most

accurate of the two phase models since it considers each phase independently and links both phases

with six conservation equations. The mass and momentum balance equations are written in

dimensionless form. The dimensionless mass and momentum balance equations are combined with

the method of characteristics and an explicit method to simulate the flow. At first, the linear

stability of the flow is investigated by disturbing the liquid flow with a small perturbation. An

improved version of the one-dimensional two-fluid model for horizontal flows is developed as

a set of non linear hyperbolic governing equations. The model accounts for the flow and

geometrical conditions effects (such as liquid viscosity, surface tension). In addition, the slope

effect of the pipe is analyzed. It is shown that, for positive values of the slope angle (upward

inclination), the slug flow becomes more probable, whereas negative values of the later

(downward inclination) induce a more stable stratified flow.

1. Introduction

2

Two-phase gas/liquid flow through a horizontal pipe displays a number of different interfacial

configurations. The knowledge of the geometric distribution of both phases under flowing

conditions is of importance. At low values of liquid flow, a stratified or stratified/wavy regime can

occur. A variation on the liquid velocity at constant gas velocity stimulates this stratified flow to

change to “intermittent” patterns, either plug or slug flow.

Theoretical prediction of the onset of slugging in horizontal or near-horizontal pipe flow has

received much attention over the last decades. Early works of Wallis and Dobson (1973) as well as

Kordyban and Ranov (1977) considered that the transition from the stratified to slug flow could be

described in terms of Kelvin-Helmholtz (K-H) instability analysis, whereby slugs arise from the

growth of infinitesimal disturbances at the interface.

Since the spacing between slugs is large enough compared to the pipe diameter the analysis can be

restricted to waves that have a large wavelength compared to the gas space. The analysis of K-H

instability is based on Milne-Thompson’s work (1968). The physical interpretation of the K-H

instability given by Eq. (1) is that the presence of the waves at the interface causes a maximum

velocity in the gas at the crest and a minimum velocity at the through. According to Bernoulli’s

equation this flow variation is associated to a minimum pressure at the liquid surface at the crest

and a maximum pressure at the trough. If the destabilizing influence of these forces is large

enough to overcome the stabilizing effect of gravity, instability occurs.

In his model, Wallis (1969) introduced wave stability conditions, predicting exponential wave

growth once the kinematic wave velocity 2

wv exceeds the dynamic wave velocity c2. This

condition of stability allowed some authors to conclude that intermittent flow can only be initiated

in supercritical flow when the bulk liquid velocity is larger than interfacial waves velocity relative

to the liquid (Taitel and Dukler, 1978). However, the experimental work of Grolman et al. (1996),

under subcritical conditions, shows that supercritical flow is not necessary for the onset of slug

flow in inclined pipe.

Taitel and Dukler (1976) proposed a model for predicting flow regime transition in horizontal and near-

horizontal gas-liquid flow. The approach, of the authors, is to assume stratified conditions and then

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determine the mechanism by which stratified flow changes to any other flow pattern. Their model for

stratified flow is one-dimensional, since both phases are treated as bulk flow. The Barnea et al.

development (1980) followed a very similar method and gave approximately the same result for

horizontal and inclined flows.

Barnea and Taitel (1992, 1994) performed a thorough stability analysis of stratified two-phase

flow. The authors claimed that two kinds of stability analysis should be conducted, i.e. interfacial

stability analysis and structural stability analysis. The interfacial stability analysis (linear and non-

linear) was used to check the stability of the interface, from which information about the later and

transition from stratified flow to other flow patterns are available. The structural stability analysis was

to check on whether the typical structure of stratified flow could be maintained, and was used to

distinguish physically stable from unstable solutions. Structural stability analysis showed that three

solutions exist, the lowest value is always stable, the intermediate one was linearly unstable, and the

largest one was nonlinearly unstable. It was also shown that the solution associated to the highest

value was usually unstable from a K-H stability analysis.

Inviscid long wavelength analysis requires a large value of the product kRe, where Re is the

Reynolds number and k is a dimensionless wave number. For a flow in a channel, the inviscid flow

assumption is not valid for very long wavelengths. Thus, Hanratty and Hershman (1961)Hanratty

and Hershman, 1961. T.J. Hanratty and A. Hershman , Initiation of roll waves. AIChE J. 1 (1961),

pp. 488–497. Full Text via CrossRef developed viscous (VLW) stability theory to describe the

initiation of roll waves in a rectangular channel. The basic assumptions were that the waves are

long enough that a hydrostatic approximation can be used to describe the pressure variation in the

liquid and that the stresses in the time varying flow can be described by making a pseudo-steady

state approximation.

Lin and Hanratty (1986) and Wu et al. (1987) applied this theory to the complicated flow that

exists in a pipe flow. They re-examine the growth of small amplitude long wavelength

disturbances. In these studies, the viscous linear stability analyses have been carried out for small

values of kh and kD. ¶ This approach differs from classical K-H linear stability theory in that

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liquid phase viscous and inertia terms are included.¶ Furthermore, the authors showed that, at

high gas velocities, slugs are initiated by the coalescence of K-H waves and not by the growth of

small disturbances.

The VLW stability theory describes transitions in horizontal air–water flows (Lin and Hanratty,

1986) at low gas velocities, even though slugs are observed to evolve by bifurcation of small

wavelength gravity waves, rather than the growth of very long wavelength waves (Fan et al.,

1993). Woods and Hanratty (1996) have argued that, in horizontal stratified flows, long

wavelength instability might be an initial step to enable smaller wavelength waves to evolve into a

slug.

Also, this theory has been strongly confirmed by Woods et al. (2000) by taking air and water

flowing in a downwardly inclined pipe. Waves are damped down at low gas velocities in an

inclined pipe, so the interfacial stress at transition is easily estimated. This study provides

observations on the appearance of long wavelength waves and their growth into a slug at a gas

velocity predicted by the viscous long wavelength analysis.

However, the VLW theory predicts very different effects of liquid viscosity. This consideration

motivated a study of the effect of liquid viscosity by Andritsos and Hanratty (1989) in horizontal

pipes. Their measurements agree with VLW theory only for low viscosities. In addition Andritsos

et al. (1989) developed a linear instability analysis applicable to high values liquid viscosity and a model to

predict pressure drop and hold-up for horizontal stratified flow in a pipe. They showed that waves in stratified

flow significantly increase the pressure drop and thus change the liquid height from the value for a smooth

interface. From this study, at large liquid viscosities the theory predicts that large amplitudes waves will be

the first instability to be observed with increasing gas velocity.

Both liquid film height and wave velocity were measured by Andritsos and Hanratty (1987),

using two parallel wire conductance probes. Air was used as the gas phase, and a water/glycerin

solution with different viscosities was used as the liquid phase. Analysis of experimental data

showed that the interfacial friction factor increased linearly with superficial gas velocity, when the

latter was larger than that needed to initiate waves. The interfacial friction factor was also affected at

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a lesser extent by liquid viscosity and liquid flow rate. An empirical correlation was proposed to

estimate the interfacial friction factor of stratified-wavy flow.

Wu et al. (1987) investigated flow pattern transitions in two-phase gas-condensate flow at high

pressure, both experimentally and theoretically. The authors conducted experiments using a gas-

condensate system in a horizontal pipe at a pressure of 75 bars. The transition boundaries between

stratified and non-stratified flow patterns were measured. It was also found that entrainment took

place in stratified flow and the gas-liquid interface was not flat, but exhibited a downward concave

configuration. The authors used both Taitel and Dukler (1976) instability model, based on the Kelvin-

Helmholtz instability, and Wallis (1969) model, based on linear stability analysis, to calculate the

transition boundary. Comparison of model predictions and experimental data showed that the

transition took place at a much higher superficial liquid velocity than predicted by the Taitel and

Dukler model. The Wallis model gave a better prediction of the transition boundary between

stratified and non-stratified flow patterns.

A simple one-dimensional three-fluid model, based on a correlation of the results of several

experiments for the simulation and analyses of vertical annular and stratified horizontal or

inclined two-phase flows, has been produced by Stevanovic et al. (1995). The general three-

fluid approach of the model ensures the testing of various interfacial transfer correlations and

the prediction of all one-dimensional flow parameters of liquid film, gas and entrained

droplets. Gorelik et al. (1999) acquired an exact analytical solution of interface configuration for

stratified two-phase flow. The solution was determined by dimensionless parameters, i.e.

holdup, fluid-wall wetted angle. Also, they showed that the constant characteristic curvature is a

good assumption.

The stability of gas-liquid stratified flow regime in horizontal annular channels has been

investigated by Akbar et al. (2003). Based on the available air-water experimental data, a simple

criterion for the prediction of conditions that lead to flow regime transition out of the stratified

wavy flow pattern is proposed. The method is an extension of the instability criterion of Taitel and

Dukler (1976) for gas-liquid flow in near-horizontal pipes.

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To understand the mechanisms involved in the evolution of waves at the transition from smooth to

wavy stratified flow regime, Lioumbas et al. (2005) suggested that the transition from a smooth to

a wavy interface can be related to the transition from laminar to turbulent flow inside the liquid

layer. The paper of Teyssedou et al. (2005) presents experimental counter-current air–water flow

data on the onset of flooding and slugging, the slug propagation velocity, the predominant slug

frequency and the average void fraction collected by using different size orifices installed at two

locations in a horizontal pipe. Teyssedou et al. (2005) observed that the mutual interaction of

waves traveling in opposite directions seems to control the behavior of the slug propagation

velocity, the slug frequency and average void fraction with the increase of gas superficial velocity.

Guo et al., (2002) investigated the effect of gas and liquid flow rates, liquid viscosity and surface

tension on the stability of the interface using a two-fluid model. They stated that surface tension !

is a stable factor of the interface such that increasing ! causes a decrease of the amplification

factor of the fastest growing waves and an increase in the wavelength. The authors found that the

non-linear stability confirms at first the conclusions reached by the linear instability, and then

gives an insight into the growth and propagation of the interfacial disturbances. The model

developed by Gu et al. (2007) is an extension of the model of Guo et al. (2002) in which the

pressure is evaluated using the local momentum balance rather than the hydrostatic approximation.

From the paper of Gu et al. (2007), both predicted results and experimental data showed that

critical liquid height is insensitive to small pipe inclinations, while at low gas velocities, critical

liquid velocity and critical wave velocity are surprisingly sensitive to small pipe inclinations.

The effect of pipe inclination on oil-water two-phase flow patterns, phase holdups, and pressure

drops were also studied by Rodriguez and Oliemans (2006). For downward inclined flow, a stable

wavy structure was observed, while for upward inclined flow, the stratified smooth flow pattern

disappears with inclination and is replaced by a stratified wavy flow pattern. Al-Wahaibi and

Angeli (2007) developed a model based on K-H stability analysis. According to this model, the

required amplitudes and lengths to initiate instability decrease as the superficial water velocity

increases for a given superficial oil velocity. At higher water velocities, any disturbance in the oil–

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water interface will become unstable. The wave growth and hydrodynamic instability criterion in

two-fluid flow was investigated by Ansari et al. (2007) using numerical simulation. The authors

found that the K-H instability criterion is correct only for long waves with small amplitude, while

the required gas velocity for the initiation of short wavelengths instability is lower than that

defined by the K-H instability relation.

Simmons and Hanratty (2001) studied the effect of small upward inclinations on the transition

from stratified to slug flow in the case of slightly inclined pipes. They showed that due to their

behavior (saw-tooth-like) the wave structure cannot be analyzed by linear theory and must

therefore be related to nonlinear phenomena. Despite the theoretical and experimental

investigations on stratified flow in inclined pipes, the issue of multiple holdups at given

operational conditions attracted a rather limited attention in the literature. Exact solutions for

steady flow with a smooth interface indicate that triple solutions are obtained in a limited range of

flow parameters for co-current up-flow and down-flow (Ullmann et al. 2003). The introduction of

multiple-solution regions on flow pattern maps of various two-phase systems shows their practical

significance, and boundaries that may be associated with flow pattern transition. Dyment et al.

(2004) obtained analytical solutions for flows in downwardly inclined ducts. This analytical model

predicts the transition between roll waves and slug regimes and gives access to all flow

characteristics without any need of closure laws concerning either the speed of propagation or the

slug length. The authors found that the onset of downward slugging is close to that observed

experimentally. In a recent paper, a nonlinear wave model developed by Johnson et al. (2009)

predicted the increase in interfacial friction for horizontal flows and flows at upward inclinations.

In this paper, predictions using VLW stability theory were particularly good at large gas flow rates

where the long wavelength instability formed roll waves. For small gas flow rates VLW stability

theory did not agree as well where the formation of roll waves was attributed to interaction

between two-dimensional waves.

To our knowledge, stratified flow in an inclined pipe has not been sufficiently investigated. The

aim of this paper is to examine transient gas-liquid flow behavior through horizontal pipe,

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including viscosity, surface tension and slope effects, using a two-fluid model combined with the

method of characteristics and linear and non-linear instabilities.

The mathematical formulation of the one-dimensional two-fluid model for horizontal flows is

first presented in which the two-phase stability problem is formulated. The linear stability

analysis with particular emphasis on the effects of flow and geometrical conditions (such

as liquid viscosity, surface tension) is then covered. A simplified non-linear analysis using

numerical simulation with particular emphasis on the effect of the inclination follows. This was done

using an explicit finite difference scheme that is based on the characteristics in a way similar to that

proposed by Crowley et al. (1992).

2. One-dimensional two-fluid model development

It is generally recognized that the original K-H criterion predicts critical gas velocities that are too

high for the onset of both waves and slugs. To obtain better agreement with experimental data,

several authors modified the criterion by introducing an empirical constant cK-H summarized as

follows:

/g K H g g gU U c (ñ ñ )gh ñ!

! " !l l

(1)

with:

cK-H = 1.0; Kelvin (1871)-Helmholtz (1868)

cK-H = 0.5; Wallis and Dobson (1973)

cK-H = 0.74; Kordyban (1977), with hg at the wave top

cK-H = 0.487; Mishima and Ishii (1980)

Wallis and Dobson (1973) obtained empirically cK-H = 0.5 with hg being the average height of the

air passage. While Mishima and Ishii (1980) used long-wave approximations in a linear stability

analysis and obtained cK-H = 0.487. Unlike the above listed authors, Kordyban (1977) takes hg at

the wave top, resulting to a different value, cK-H = 0.74.

Taitel and Dukler (1976) extend the K-H theory to pipes of circular cross section and finite

amplitude waves. The authors suggest that the instability occurs when:

9

dh

dA

AgdhU

g

gg

gs

l

l

!

"!! )cos()()/1(

##$ (2)

In the one-dimensional wave model developed by Wallis (1969), the instability results from the

interaction between kinematic and dynamic waves. This occurs when the kinematic velocity wv is

higher than the dynamic wave velocity c expressed as:

2 2

wv > c (3)

This criterion is the basis of the present analysis.

2.1. Interfacial stability of stratified gas-liquid flow: Two-fluid model

The “two-fluid model” treats the fluids separately as if each one flows in its own channel within the

pipe. The model consists of the continuity and momentum phase equations leading to a set of four

non-linear partial differential equations. By assuming isothermal conditions, the energy equation is not

required.

The flow depicted in Fig. 1 is assumed to be co-current gas-liquid in a pipe of diameter d and

liquid depth h. The interface length as well as the chords on the pipe circumference that are in

contact with gas and liquid are given respectively by Si, Sg and Sl. Ag and Al are the gas and liquid

cross sections.

Assuming incompressible flow, no mass transfer between the gas and the liquid (fully immiscible

liquids), hydrostatic pressure distribution in the y-direction and distribution coefficients nearly one

(thick liquid layer assumption), the continuity and momentum phase conservation equations are given

respectively as:

0)()(=

!

!+

x

UA

Dt

AD j

j

jj (4)

x

AP

x

PAgASS

Dt

UDA

j

ij

jj

jjiijwj

jj

jj!

!+

!

!""±"= #$%%$ sin

)( (5)

10

Where x

UtDt

Dj

j

!

!+

!

!= and the subscript j stands for liquid l=j and for gas gj = . The upper

sign of “ ± ” corresponds to the liquid phase.

The average pressure jP differs from the corresponding value at the interface ij

P since the

pressure of each phase varies due to gravity. The average pressure of each phase, in terms

of its pressure level at the liquid-gas interface, y= h, is given by

( ) ( )( )

0

cos ( ) cos

h xj j j ij

ij j j

A P A P hP g h y bdy g

x x x x

! !" # " #

! !

! !$ %= + & = +' (! !) (6)

where b=b(y) is the interfacial perimeter.

Substituting the above relation, Eq. (6), into Eq. (5) one can obtain the combined momentum

equation for two-phase flow as follows:

x

hgA

x

PAgASS

Dt

UDA jj

ij

jjjiijwj

jj

jj!

!+

!

!""±"= #$#$%%$ cossin

)( (7)

Considering ( )j jA A h= , the mass conservation equation reduces to:

0'

=!

!±

x

U

A

A

Dt

hD j

j

jj (8)

where '

j jA dA / dh= . The ratio '

j jA /A can be interpreted as an average or equivalent height and is

given for the liquid phase by:

1

' 2

1 cos (2 1)(2 1)

4 1 (2 1)i

A A hh h

A S h

! "# $" "% &= = + " <% &" "' (

l l

l

%%

%

with h h / d=% .

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Fig. 1. Definition of parameters for stratified pipe flow.

A one dimensional wave model flow is considered (Crowley et al. 1992). This model allows

analyzing the influence of a perturbation on flow parameters, for horizontal or nearly-horizontal

pipes. The momentum equation, Eq. (7), can be combined into a single equation as follows

Fx

PP

x

hg

Dt

UD

Dt

UD igi

g

g

g =!

"!+

!

!"+"

)(cos)(

)()( l

l

lll

l#$$$$ (9)

where F is expressed as:

( )1 1

sinwg gw

i i g

g g

ô Sô SF = + +ô S + ñ ñ g è

A A A A

! "# # #$ %$ %

& '

l l

l

l l

In this model, the continuity of the normal stresses (pressure) is balanced only with the interfacial

tension

1 2

1 1

ig iP P

r r!" #

$ = +% &' (

l (10)

where r1 and r2 are the interface principal radii of curvature and ! is the air-water surface tension.

The curvature radius r1 is evaluated in the plane that contains the x-axis, while r2 is the

curvature radius on the plane perpendicular to this axis (Fig. 2). The interface is assumed to be flat

in the cross section and wavy in the axial direction. This is the one-dimensional wave

approximation. Therefore, r1 is the curvature radius of a plane curve and r2 is infinite. That is,

2

2

32 2

1

1

1

h

x

rh

x

!

!=

" #!$ %+& '( )& '!* +, -

, 2

10

r

= (11)

Fig. 2. Curvature radius on the liquid-gas interface (a) r1 and (b) r2.

12

Combining Eqs. (10) and (11), the pressure jump across the interface can be expressed as

( )2

2

32 2

1

ig i

hP P x

x xh

x

!

" #$ $%$ $% & % $ $%= ' (

% % $ $) *%+ ,+- .$ $/ 0- .%1 2$ $3 4 56

l (12)

where ! is considered as constant.

In order to use the one-dimensional analysis, the interfacial waves must be of infinitesimal

amplitude. The wavelength! can be shorter than, equal to or greater than the pipe diameter d.

Waves are considered as short waves for ! << d and as long waves for ! >> d, respectively. As

Barnea and Taitel (1994a) pointed out that short waves cause a pebble interface rather than a

transition. On the other hand, long waves may produce a flow pattern transition. Therefore, this

analysis will be restricted to long waves.

For the long wave approximation ( h / x 1! ! << ), Eq. (12) reduces to

( ) 3

3

ig iP P h

x x!

" # "=

" "

l (13)

Substituting Eq. (13) into Eq. (9), the combined momentum equation reduces to

( )3

g g

g 3gcosg g g

U UU U h h+ U U ñ ñ è = F

t t x x x x! ! ! ! "

# ## # # #$ $ + $ $

# # # # # #

l l

l l l l (14)

Setting F = 0 yields the equilibrium solutions. The solution for the equilibrium liquid level

requires expressions for w!

l,wg! and

i! in terms of the velocities of the two fluids. The wall shear

stressesw!

land

wg! are conventionally expressed in terms of the corresponding friction factors

wf

land wgf .

2

;2

ww w

f U DUf

!"

# $%

&' (

= = ) *+ ,

l l l l l

l l

l

2

;2

wg g g g g

wg wg

g

f U D Uf

!"

# $%

&' (

= = ) *) *+ ,

13

For gas-liquid flow, Taitel (1977) stated that this is a good approximation for horizontal pipes

(gravity independent equilibrium level) but not for inclined pipes. The coefficients χ and the

exponent η are equal to 0.046 and 0.2 for turbulent flow and 16 and 1.0 for laminar flow,

respectively. In this study, interfacial shear stress is expressed as follows:

0

i i s! ! != + (15)

where

( )0

2

i g g g

i

f U U U U!"

# #=

l l

According to Barnea and Taitel (1992)

fi=0.014 if fwg<0.014

fi= fwg. if fwg>0.014

and

( )2

s f g s

hU U C

x! "

#= $

#l

(16)

f! !=l if g

U U>l

f g! != if gU U<

l

Note that 0

i! considers only the interfacial friction along S, when the interface is smooth. The wave

effects are incorporated into the model via s! and the parameter Cs must be evaluated empirically (Cs

>0). Introducing the s! definition, Eq. (16), into Eq. (14), the combined momentum equation

acquires its final form:

3

g g

g 3g g

U UU U h h+ U U G = F

t t x x x x! ! ! ! "

# ## # # #$ $ + $

# # # # # #

l l

l l l (17)

where

( ) ( )2 1 1

cosg f g s

g

G g ñ ñ è U U CA A

!" #

= $ $ $ +% &% &' (

l l

l (18)

And F takes a final form as

14

( )0 1 1sin

g g

i i g

g g

SSF = S g

A A A A

!!! " " #

$ %& + + + & &' (' (

) *

l l

l

l l

(19)

The water equilibrium level, a solution of F = 0, is independent of the waviness characteristics of

the interface, and only the quasi-steady term of the interfacial shear stress is included.

3. Linear Stability Analysis

One of the primary ways for determining the flow regime transition point from stratified wavy

flow to slug flow is with a linear perturbation analysis. By incorporating a linear perturbation

analysis, the flow instability of the system can be determined. The proces s o f det er min ing

th is int er fa ci a l instab i l it y is given in Barnea et al. (1993) and Brauner et al. (1992)

papers. First, the continuity and momentum equations are written and constitutive relations are

applied for each phase assuming that the flow is one-dimensional and both phases are

incompressible. In order to set the conditions of the interfacial waves that lead to instabilities the

behavior of very small perturbations can be examined using the perturbed flow equations. This

procedure is known as a linear perturbation analysis. In order to obtain the perturbed flow

equations, first the three basic flow variables,Ul, gU and h are written as:

' ' '; ; ; ( , , ) 0g g g gU U u U U u h h h F U U h= + = + = + =l l l l

where

'/u U

l l , '/

g gu U ,

'/ 1.h h <<

It is assumed that the perturbations are very small compared to the mean variable values, therefore

the higher order (nonlinear) terms of the perturbed equations may be dropped.

The linearization of the continuity equation Eq. (8) is straightforward, and results in

'' '

0'

A uh h+U + =

t x xA

!! !

! ! !

l l

l

l (20)

and :

'' '

0g g

g'

g

A uh h+U =

t x xA

!! !"

! ! ! (21)

For the momentum equation:

15

' '' ' ' 3 'g

g 3

g

g g

u uu u h hñ ñ + ñ U + ñ U G = F

t t x x x x!

" "" " " "# + # "

" " " " " "

l l

l l l

(22)

where the quantity F! is made up of the following contributions owing to each perturbation:

,, , gg

U UgU h U h

F F FF

U U h

! "! "# # #! "# = + +$ %$ % $ %$ % $ %# # #& '& ' & ' l

l

l

The velocity perturbations 'u

l and '

gu can be eliminated from the above equations by

differentiating Eq. (22) with respect to x and then substituting Eqs. (20) and (21) in the

resulting expression. Grouping terms and multiplying all the equation by '/A A!

l lresults in

24 ' 2 ' 2 ' 2 '

2

4 2 2' '

'

', , ,g

g

g g g g g

g g g

g

U U g gU h UU h

U A U A AA Ah h h hU G U

x x t x tA A AA A

A AF F F h FU U

xh U A U UA

! ! !" ! ! !

# $ # $ # $% % % %& '+ + ( + + + + =& ' & '

% % % % %& ' & ' & ') * ) *) *

# $+ ,+ , + ,% % % % %+ ,& '( + ( +- .- . - .- . - . - .- .& ' %% % % %/ 0 / 0 / 0/ 0) *ll

l l ll l

l l l l l

l l

l l

l

l ll

'

, ,g

g

g gh U h

A F hU

tA U

# $+ ,% %& '( - .- .& ' %%/ 0) *l

l

(23)

Substituting a monochromatic wave for the perturbed water level:

' ( )i t kxh h e

! "=

into the above eq. (23) yields

2

4 2 2 2

' '

', , ,,

2

gg g

g g g g g

g g g

g g

U U g g g gU h U hU h

U A U A AA Ak U G k U k

A A AA A

A A AF F F F FU U ik U

h U A U U A UA

! ! !" ! ! # ! #

$ % $ % $ %& '+ + ( + + ( + =& ' & '& ' & ' & ') * ) *) *

$ %+ , + ,+ , + ,- - - - -+ ,& '( ( + ( + (. / . /. / . /. / . / . /. / .& '- - - - -0 1 0 1 0 10 1 0 1) *ll

l l ll l

l l l l l

l l

l l l

l

l ll ,U h

i#$ %& '

/& ') *l

(24-a)

Substituting A A!=l l

and g gA A!= into Eq. (24-a), yields

22

4 2 2

' '

, , ,

2

s gs gs g s

g g g g g

g g g

s gsU U U U

U A UA U UAk G k k

A A

F F Fik i

U U! !

" " "" " "# $ $

! ! ! ! ! !

$!

% & % & % &' (+ + ) + + ) + =' ( ' (' ( ' ( ' (* + * +* +

% &% & , -, -, -. . .' (' () ) + ) / 0/ 0/ 0/ 0 / 0 / 0' (' (. . .1 2 1 2 1 2* + * +l l l

ll l l l l l

l l ll l

l l

(24-b)

where ! stands for the phase holdup, s

Ul

and gsU liquid and gas superficial velocities respectively.

Solving for the angular frequency,! , the following dispersion relation is found:

16

2 2 42( ) 0ak bi ck Dk eki! !" " + " " = (25)

where

, ,

22

' '

,

1 1;

2

1;

1

gs g s

s gs

g g

g gU U

g g

g

g

gU U

UU F Fa b

U U

U AU A Ac G D

A A

Fe and

! !

""

" "! !

"" #

" "! !

"""

" ! ! !

$ %$ % & '& '( () *) *= + = + , -, -, - , -) *( () * . / . /0 1 0 1

$ %) *= + + =) *0 1

$ %& '() *= + = +, -, -) *(. /0 1

l l

l

l l

l l

ll l

l l l

l

l l

The derivatives can be calculated either explicitly or numerically. Explicit derivatives are obtained

by differentiating the RHS of Eq. (19) (with constant friction factor fi) to obtain:

23

,

1 1

gs g

gss sii g i

s g gU

Uf S U USFf S

U A!

""

! ! ! ! !!

# $% &% &% &'( )= * + * ++ ,+ ,+ ,+ , + ,+ ,' ( )- . - .- ./ 0

l l l l l

l l l ll

(26-a)

23

,

1 1

s

g g g gs gs sii g i

g g g ggU

f S U U USFf S

U A!

""

! ! ! ! !!

# $% & % &% &' ( )= * + * ++ , + ,+ ,+ , + ,+ ,( )'- . - .- ./ 0l l

l

l l

(26-b)

22

2 22

, , ,

22 2'

2 22

2

1

22

s gs gs g s

gs gss s sii g i

s g g gU U U U

g g g gs gs s sii g i

gg g

U UU f S U USF F Ff S

U U A

f S U U U S USf S f

A A

! !

""

! ! ! ! !! !

"" "

! !! !

# $% & % &% &% &' ' ' ( )= * * * *+ , + ,+ ,+ ,+ , + , + , + ,( )' ' '- . - . - . - ./ 0

# $% &( )* * * ++ ,

+ ,( )- ./ 0

l l l

l l l l l l

l l l ll l

l l l

l l

l

32

22

2 22

1 1

2

g g g gs gs sii g i

g gg g

f S U U USf S

A

!

""

! ! ! !! !

# $% & % &( )* * * * ++ , + ,

+ , + ,( )- . - ./ 0

l

l

l l

(26-c)

The numerical approach evaluates the derivatives by perturbing !l sU

land gs

U in the RHS of Eq.

(18) by small quantities (say ± 510-2 %), where the dimensional geometrical variables are (see

Fig.1)

17

2 1

2 2

/ , 2 1

1 , cos , ( )

/ 4,4

i g g

g g i g g

h h d H h

S d H S d H S d S

A d S S H A d A

!

!

"

= = "

= " = = "

# $% &= " = "' () * + ,

l

The behavior of a disturbance, whether it amplifies or decays, determine the stability for a given

mean flow condition. The solution for ! is

2 2 2 4( ) ( ) ( 2 )ak bi a c k b Dk ek abk i! = " ± " " + + " (27)

The steady-state solution is unstable whenever the imaginary part of ! in Eq. (25), namely I

! , is

negative, leading to exponential growth of the perturbed water level h’. When I

! is positive, the

interface is stable. The neutral stability condition is achieved when 0I

! = . IntroducingR Ii! ! != + ,

into Eq. (25) and allowing I

! to approach zero yields:

2 2 42 0

R Rak ck Dk! !" + " = (28)

2 0R

b ek! " = (29)

Equations (28) and (29) define the stability line. From Eq. (29) it follows that

2R

ek

b! = (30)

and

,

, ,

2

s gs

gs g s

U URv

s gsU U

F

eC

k bF F

U U! !

!"

# $%& '%( )

= = =* +# $# $% %, -. & '& ' & ', -% %( ) ( )/ 0

l

l l

l

l

(31)

where Cv is the wave velocity at the onset of instability. The criterion for stability can be obtained

by substituting R

! from Eq. (30) into Eq. (28). This yield

( )2 2 2( ) 0

vc a a c Dk! ! ! ! < (32)

when inviscid flow is considered, Eq. (28) reduces to a simpler form

( ) }( )

2

4

2gcos

g gg g g

inviscid g

g g i g

U U ñ ñU U AC ñ ñ è k

S ñ

! " ! "#

! " ! " ! " "

$+ %&= ± $ + $'

+ &(

l ll l l

l

l l l

(33)

18

The amplification factor vanishes as soon the term inside the square root is zero or positive.

Otherwise, two conjugate solutions forI

! exist. The second solutionI

!" is the one that causes

instability. The critical wave velocity in this case, Civ equals to:

g g g

i

g g

U UC

! " ! "

! " ! "

+=

+

l l l

l l

(34)

and can be interpreted as an average velocity weighted by the fluid densities.

Thus the general means viscous stability criterion (32) can be expressed as:

( )( )

( )( )

2 2

2 4

2

1 1gcos 0

g g f g

v inviscid g s

gg i

U U ñ ñ U UAC C ñ ñ è k C A

A Añ S

!"

!# # !

$ $ % &' ($ = $ $ + + $ <) *+ , ) *

- .

l l l

l

ll

(35-a)

And in a simplified form:

0

VKH

U g s

IKH

J J J J Jµ !+ " + + <64444744448

1442443 (35-b)

The second up to the fourth terms on the LHS of Eq. (35-b) comprise the stability criterion for the

inviscid analysis, where the viscous effects are neglected. This is the well-known K-H instability

of the interface for one-dimensional flow. The first term is the additional effect of shear stress,

which tends to amplify any disturbance on the interface. This term is related to the difference between

the wave velocities predicted by the VKH and IKH theories. The fourth term, J! which is the

contribution of the surface tension, is the only term that depends on the wavelength. For long

waves ( 0k ! ) the terms approaches zero and does not affect the neutral stability criterion. The

sheltering effect is isolated in the last term, Js and represents the other source of instability.

In order to check the accuracy of the two-fluid model, the latter is compared with exact solutions

for potential and viscous flows. For simplicity, and considering the inherent nature of the two-fluid

model, the exact solutions are evaluated for the case of wide parallel plates as shown in Fig. 3.

The potential flow solution by Lamb (1932) states that the wave celerity is given by

( ) ( )2

2

1 g ggg g

Lamb

f g f f f g

U U ñ ññ ñUU kC

T T kñ ñ ñ T T

!! "

!

#$ % #= + ± + #& '

& '( )

l lll l

l l

(36)

19

where g

f

gT T

!!! = +l

l

And tanh( ); tanh( )g gT kh T kh= =l l

The flow is stable whenever

( ) ( )2

20

g g g

f g f f

U U ñ ñ g ñ ñ k

ñ T T kñ ñ

!" "" " <

l l l

l

(37)

is satisfied and the amplification factor is

( ) ( )2

2

g g g

I

f g f f

U U ñ ñ ñ ñ kk

ñ T T kñ ñ

!"

# ## = # #

l l l

l

(38)

In contrast with the two-fluid model, the gravity contribution in Eq. (37) is wavelength dependent

and the interfacial tension term is not simply proportional to k2. The gravity term

dominates for long waves ( 0k ! or d! >> ) and the interfacial term is important in the

short wave region ( kh 1>>l

or d! << ) . However, in t he l imit of long waves, both

models have almost the same r esul ts ( tanh(kh ) kh!l l

as kh 1<<l

and

/ /f g gk h h! ! !" +l l

while for short waves ( tanh(kh ) 1!l

as kh 1>>l

. The models differ, as

depicted in Fig. 3. Both models give the same results for wavelengths equal to or larger than the

pipe diameter.

Fig. 3. Comparison of the Lamb (1932) potential flow solution and the two-fluid model (IKH).

20

3.1. Computational procedure

It is easy to solve the quadratic equation (25). The main components of the calculation procedure

are summarized as follows:

Input the required known parameters i.e. geometrical parameters gas and liquid flow rates.

The fully developed stratified gas-liquid flow is solved in order to determine , gU U and hl

.

The coefficients of eq. (25) are estimated.

Finally, solve the eq. (25).

The two phases air-water used in this study were considered. The properties of water and air are

taken at standard atmospheric pressure.

3.2. Results

An interesting result of this study is the effect of the viscosity on the amplification factorI

!" .

From Fig. 4, it is seen that there is a maximum value for each curve. This value stands for the

amplification factor of the fastest growing waves, ( )maxI

!" .The wavelength of the fastest growing

waves is represented bymax! . ( )

maxI!" decreases very rapidly as the liquid viscosity decreases.

The points at which I

!" becomes zero are defined as the neutral stability points. The latter is

reached, as depicted in Fig. 4, for values smaller than 7.97 10-4 Pa.s. It is generally known that

increasing the liquid viscosity induces a more stable flow. Also, according to the traditional

theory, any flow will be stable when the liquid viscosity is high enough. In general, the viscosity

has a dual effect on the flow: increasing viscosity can increase instability due to velocity profiles

differences at the interface of the two layers, and at the same time it helps to dissipate the energy

that causes instability (Yih, 1967). However, as illustrated in Fig. 4, with increasing liquid

viscosity, I

!" increases, whilemax! decreases.

21

Fig. 4. Effect of liquid viscosity on the amplification factor

(Uℓs=0.5 m/s, Ugs=5 m/s, σ = 0.072 N/m, d=0.05 m and θ=0°)

Fig. 5 exhibits the effects of surface tension on the amplification factor. With increasing surface

tension, I

!" decreases whenmax! increases. As shown in Fig. 5, the surface tension has a great

influence on I

!" only within a short wavelength regime. Beyond a value of 1 for the ratio d/! ,

there are no explicit differences between the curves. To our knowledge, no published study has

quantified the particular effect of surface tension without changing other fluid properties. It is

believed that an increase in surface tension tends to reduce the unstable growth of the short

waves at the interface. As shown in Figs. (4-5), the results are in agreement with theoretical

results of Guo et al. (2002).

22

Fig. 5. Effect of surface tension on the amplification factor

(Uℓs=0.5 m/s, Ugs=5 m/s, µ = 7.9 10-4Pa s, d=0.05 m and θ=0°).

4. Non-Linear Stability Analysis of the Stratified/Non-Stratified Transition

The stratified/non-stratified flow pattern transition has been explored in the previous section, using

the K-H linear stability analysis. The linear stability analysis can be only used to distinguish

whether an interface is stable or not. Once the interface is unstable, this theory cannot

distinguish between a wavy interface and transition to a non-stratified flow pattern. In order to

explore the flow response to finite interfacial disturbances, a non-linear analysis has been carried

out using numerical simulation.

3

30

U U h hñ + ñ U L E =

t x x x!

" " " "+ # +

" " " "

l l

l l l (39)

where

( )2

2

3cos

'

g g gs

g

A AL g ñ ñ è U

A!= " " l

l

( )1 1

sing g

i i g

g g

SSE = S g

A A A A

!!! " " #

$ %& & + + &' (' (

) *

l l

l

l l

As a further approximation, the surface tension will be dropped ( 0! = ). This is physically

accepted in the long wave limit since the interfacial waves have a large wavelength compared to

23

the liquid film thickness in the range of parameters studied 0k ! . Therefore, the system of

equations can be reduced to:

0'

A Uh h+U + =t x A x

!! !

! ! !

l l

l

l

(40)

0U U h+U L E

t x x

! ! !+ + =

! ! !

l l

l (41)

4.1. Numerical method

The method of characteristics is a standard approach for solving partial differential equations

(PDE) of the hyperbolic type (i.e. those with real and finite wave velocities). The technique is to

find the directions in the (x-t) plane along which the formulation can be recast as ordinary

differential equations (ODE).

This method is often used to solve transient flow problems. It is well known that the solution

proceeds numerically along the paths followed by waves in the physical plane. This method has

the advantage of leading to a clear understanding of the physical implications of the numerical

procedures. However, the technique has the disadvantage of being stable only when relatively

small time increments are used in the stepwise solution. The method was originally developed to

be used with a characteristic net, but it is now much more common to use fixed grids where results

are derived at predetermined points in the physical plane.

Unlike other finite difference schemes, the method of characteristics reduces the numerical

diffusion and allows an accurate simulation of the wave evolution.

Eq. (40) and (41) can be expressed as a standard non-linear hyperbolic governing equation

0X X

N Mt y

! !+ + =

! ! (42)

Where:

[ ],T

X h U=l

[ ]0,T

M E=

24

'/U A A

NL U

! "= # $% &

l l l

l

The two characteristics values of matrix N are:

' 1/ 2

1

' 1/ 2

2

( / )

( / )

U L A A

U L A A

!

!

= "

= +

l l l

l l l

1! is the characteristic with the slower velocity while

2! is the characteristic with the faster velocity.

1! can be positive or negative. It is negative for subcritical flow and positive for supercritical flow.

Also, 1! and

2! are corresponding to the two propagation velocities of disturbances

respectively.

Let us consider a nonsingular matrix K (Quarteroni et al., 2000), such that

1N K K

!= " (43)

where:

1

2

0

0

!

!

" #$ = % &

' ( is the diagonal matrix of engeinvalues of matrix N

K is the matrix for characteristic vector of matrix N

1/ 1

1/ 1

BK

B

! "= # $%& '

'

AB

A L

! "= # $% &

l

l

Following the gas-liquid theory (Barnea and Taitel, 1994b), after multiplying Eq. (42) by the

matrix for characteristic vector K on the left-hand side, a set of characteristics equations was

obtained:

1 1 10

U Uh hB + BE along

t t t x! ! !

" "" " # $+ + + =% &" " " "' (

l l

(44)

2 2 20

U Uh hB + BE along

t t t x! ! !

" "" " # $+ + + =% &" " " "' (

l l

4.2. Explicit finite difference scheme

The ODE set (39) is discretized by space-time explicit finite difference scheme (Fig.6) as follows:

25

1t t

i ih hh

t t

+!"

=" #

(45)

( ) ( )1t t

i iU UU

t t

+!"

=" #

l ll

( )

( )

1

1

0

0

t tii i i

i

t tii i i

h h ifh x

xh h if

x

!!

!!

!

"

+

#" $%& %'

= (& % " <

%')

(46)

( ) ( ) ( )( )

( ) ( )( )

1

1

0

0

t tiii i

it ti

ii i

U U ifU x

xU U if

x

!!

!!

!

"

+

#" $%& %'

= (& % " <

%')

l l

l

l l

Finally, once the values of h et Ul are known at time step t, they can be obtained at time step

t t+ ! :

If 10! " :

( ) ( ) ( ) ( )( ) ( )

1 21 21

1 1( )

2 2

t ti i it t t ti i

i i i i i i

Bt th h h h U U

x x

! !! !+

" "

# $++ % %& ' # $= " " " "& '% %

l l (47)

( ) ( )( ) ( ) ( ) ( )

( ) ( )1 1 21 2

1 1( )

2 2

t t t ti it ti i

l l i i ii i i i

i

t tU U h h U U tE

B x x

! !! !+

" "

# $++ % %& ' # $= " " " " " %& '% %

l l (48)

If 10! < :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

1

1 1 2 1 2 1

1 2 1 21 1

2

2

t t t t t

i i i i ii i i i

t t ti

i i i i i i i

th h h h h

x

B tU U U

x

! ! ! !

! ! ! !

+

+ "

+ "

# $ %$ %= " " " "& '& '## $ %$ %" " + + "& '& '#

l l l

(49)

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

1

1 1 2 1 2 1

1 2 1 21 1

2

2

t t t t t

i i ii i i i i i

i

t t t

ii i i i i i i

tU U h h h

B x

tU U U tE

x

! ! ! !

! ! ! !

+

+ "

+ "

# $ %$ %= " " " " "& '& '#

# $ %$ %" + " " " #& '& '#

l l

l l l

(50)

The stability of the above difference scheme may be guaranteed by ensuring that the Courant-

Friedrichs-Lewy (CFL) stability criterion is observed. This requires that the ratio /t x! ! of the

grid pattern shall not exceed the greatest absolute value of the characteristic.

26

maxi

ÄxÄt

!"

Fig. 6. Numerical domain of dependence spatial and temporal discretization.

4.3. Boundary conditions

The simulation procedure starts with the initial equilibrium conditions. Then, a finite disturbance is

superimposed to the steady-state values of liquid film thickness. When the flow is supercritical

(10! " ), no wave reaches the entrance at x = 0 is the initial value at t = 0. The solution domain is

considered semi-infinitely long, such that boundary conditions are required only for x = 0. For subcritical

flow (10! < ) the upstream wave is reflected at the inlet. Therefore, the boundary condition has to be

added to the equation. In this case a constant known flow rate Qg is used.

4.4. Results and discussion

In order to apply the K-H criterion, the equilibrium liquid level h (or liquid hold up) is required.

For that purpose, a circular pipe diameter is considered with liquid flowing at the bottom of the

pipe and the gas concurrently with it. The effect of a variation in the liquid flow velocity at the

entrance is studied while maintaining a constant gas velocity. The variation of the liquid velocity at

the inlet is obtained by means of using a linear opening valve law such as: ! "ls

U = + t , where:

! and! are constants.

27

The results presented in Fig.7 are also compared to those obtained by Taitel and Dukler (1978); a

good agreement is observed.

Fig. 7. Equilibrium levels by using linear opening law of a valve.

4.4.1. Effect of small disturbance on the stability of the flow

The K-H linear theory has been used to predict the onset of slugging in a horizontal duct. However

the stability analysis is based on one dimensional Bernoulli equation Taitel and Dukler (1976). The

two phase interface is perturbed by means of small disturbances and the decay/growth of a

perturbation is examined.

Initially the film thickness is constant. A liquid height disturbance is initiated at a location of the

pipe (Fig. 8). At first, it is noted that an initial disturbance generates two waves going away one

from each other (Fig. 9a). The wave traveling upstream decays with time. However if it reaches

the inlet of the pipe this wave may be reflected and then travels downstream. Hence, this wave

evolves like the downstream one. In fact, the growing of this ‘’new’’ disturbance may be

responsible of the onset of slugging. Then for many couples of liquid and gas non-dimensional

superficial velocities ( ( )0.5

j js j gj U / gd( )! ! != "l

), the downstream wave will growth (Fig. 9a

right), or decay (Fig. 9b).

28

Fig. 8. Schematic of two phase flow submitted to perturbation.

Fig. 9a. Evolution of the interfacial non-linear waves

(α=0.1; jℓ=0.2; jg=1.05, air-water, d=0.05 m).

29

Fig. 9b. Evolution of the interfacial non-linear waves

(α=0.1; jℓ=0.17; jg=0.2, air-water, d=0.05 m).

As shown in Fig. 10a, the wave at first grows to reach a maximum amplitude before decaying.

This is due to the fact that, initially, the kinematic wave velocity is greater than the dynamic one

(vw2>c2) up to the point where the interfacial friction acting on the opposite will cause a damping

of the wave. It is reported that, for some values of liquid and gas flow rates, the perturbation

amplitude increases at the onset of the latter, before damping down (Fig. 10b). This is the result of

the interaction of the kinematic and dynamic velocities. This is observed once the value of the

kinematic velocity vw becomes smaller than the value of the dynamic wave velocity c expressed

that is v cw

2 2< .

Fig. 10a. Evolution of the interfacial non-linear waves

(α=0.1; jℓ=0.106; jg=0.1, air-water, d=0.05 m).

30

Fig. 10b. Increase and decrease of wave amplitude

(α=0.1; jℓ=0.106; jg=0.1, air-water, d=0.05 m).

4.4.2 Slope effect on the stability of the stratified two phase flow

In near-horizontal pipes, stratified flow, slug flow, annular flow and dispersed bubble flow may exist

for different fluid properties (density, viscosity, surface tension), pipe geometry (pipe diameter,

inclination angle) and operational conditions (flow rates, pressure, temperature).

In the following figures, the effect of slightly upward and downward inclinations on flow stability

is examined. The stratified-intermittent transition boundary is very sensitive to the inclination

angle even less than 1°.

In Fig. 11 for upward inclination, the flow is still stable at least for ( 0.4! °" ). But if we increase

the pipe inclination, the stratified flow becomes unstable at first then stable again downstream

( 0.5! °" ). For greater inclination ( °! 6.0" ), the flow becomes unstable: which means that the

stratified smooth flow is not seen for this angle. Since, initially vw2< c2, for 0.6! °

= vw increases

faster than c, the criterion (3) is verified, and the flow becomes unstable. Therefore, positive slopes

destabilize the stratified flow, and transition to slug flow can occur.

31

Fig. 11. Effect of upward inclination on the stability of the flow

(α=0.1; jℓ=0.086; jg=0.2, air-water, d=0.05 m).

In downward inclination cases (Figs. 12a-12b) the liquid moves more rapidly and has a lower level

in the pipe owing to downward gravity forces. As a result higher gas and liquid flow rates are

needed to cause transition from stratified to slug flow. In contrast with upward inclination and

horizontal flow cases where waves are generated from the action of the gas on the interface, for

downward flow the instability takes place as result of the presence of gravity even if the gas flow

rates are negligible (Fig.13).

Fig. 12a. Effect of downward inclination on the stability of the flow

32

(α=0.1; jℓ=0.086; jg=0.2, air-water, d=0.05 m).

Fig. 12b. Effect of downward inclination on the stability of the flow

(α=0.1; jℓ=0.118; jg=0.2, air-water, d=0.05 m).

Fig. 13. Effect of downward inclination on the stability of the flow for small gas flow rate.

(α=0.1, jℓ=0.105; jg=0.1, air-water, d=0.05 m,).

In order to further enhance the confidence level in the results of the model, transition lines for

stratified/non stratified boundary were developed for different pipe inclinations.

Fig. 14(a-b) and Fig. 15(a-b) compares the calculated critical liquid height at neutral stability and

critical superficial liquid velocity with both experimental and calculated data of Gu et al. (2007),

33

the results from the long wave viscous analysis of Lin and Hanratty (1986), Barnea et al. (1993)

and Woods (1999). Based on all the comparisons presented, in general, the agreement between

the model prediction for stratified to non-stratified transition boundary and all the data sets is

good.

It should be noted that in a case of downward flow only a slight effect of the angle of inclination is

observed on critical liquid height (Fig. 14b). While an inspection of Fig.15-b suggests a great

sensitivity of flow pattern to the angle of inclination at low gas velocities. Thus at low gas

velocities, the critical liquid flow in a pipe with 0.5! = " ° is approximately seven times the value

for a horizontal pipe. This shows that at low superficial gas velocities the gravitational force plays a

dominant role in driving the liquid phase to flow in a downward inclined pipe. With an increase in

superficial gas velocity, the importance of the gravitational force decreases. At USG > 6 m/s, the

transition is weakly sensitive to declination, suggesting that the inertial effects of the gas flow

overcome the stabilizing effects of gravity.

(a)

34

(b)

Fig. 14: Neutral stability predictions-Critical liquid height. (a) air-water, d=0.05m, 0.8! = " ° ;

(b) air-water d=0.05m, horizontal and downward inclination.

(a)

35

(b)

Fig. 15: Neutral stability predictions-Critical liquid velocity. (a) air-water d=0.05m, 0.8! = " ° ;

(b) air-water, d=0.0763m, horizontal and downward inclination 0.5! = " ° .

In Fig.16, the flow pattern map is presented for various upward pipe inclinations. Both sets of

results indicate that the stratified region shrinks in size to a very small bell-shaped area as the

upward inclination angle increases. These findings seem to be consistent with the Barnea et al.

(1980) results. For up flow, conversely of downward inclined flow where stratified flow is the

dominant flow regime, upward inclination cause intermittent flow to take place over a much wider

range of flow conditions. As the angle of inclination increases, the region of stratified flow

decreases and finally completely disappears. Therefore, the stratified/non stratified transition

boundary is very sensitive to upward inclination even if 1! < ° .

36

Fig. 16: Neutral stability predictions-Critical liquid velocity, horizontal and upward inclinations.

(air-water, d=0.025m).

5. Concluding remarks

An analysis on the stability of the governing differential equations for area averaged one-

dimensional two-fluid model is presented. Both linear and non linear analyses are included.

According to the linear analysis approach the followings holds:

• The amplification factor ( )maxI

!" decreases very rapidly as the liquid viscosity decreases.

• Surface tension is a stable factor of the interface. A rise in the surface tension value,

leads to a decrease of I

!" whenmax! increases.

Non-linear analyses reveal that the stratified/non-stratified transition must be addressed with the

complete two-fluid model. Stratified flow can exist in horizontal flow as well as in downward and

upward inclined flow. In downward inclinations stratified flow is the dominant flow regime, while

in upward flow it occurs over a smaller range of flow rates, depending on the inclination angle. As

the angle of inclination increases, the region of stratified flow decreases and finally completely

disappears.

37

Moreover, the model gives reasonable predictions for pipe inclination angles, even though it was

validated only for small angles. Thus, the negative slope tends to prevent the onset of slugs in the

flow by a damping effect in wave amplitude. The opposite effect was observed for the positive

slope, for which the upward inclination destabilizes the stratified flow and transition to slug flow

can occur.

Nomenclature

Aj surface occupied by phase j ,m2

c dynamic wave velocity, m/s

d diameter of the duct, m

f friction factor

g gravitational acceleration, m/s2

h height of phase, m

jj non-dimensional superficial

velocity of phase j

38

k wave number, m-1

P pressure, Pa

Si length of the interface, m

Sj perimeter of pipe cross section

occupied by phase j, m

Uj velocity of phase j, m/s

Uℓs liquid superficial velocity, m/s

Ugs gas superficial velocity, m/s

vw kinematic wave velocity, m/s

x flow direction, m

y vertical to flow direction, m

Subscripts

j=ℓ for liquid

j=g for gas

i interface

Greek letters

α void fraction

λ wavelength, m

µ dynamic viscosity, Pa s

θ pipe inclination, degrees

ρj density of phase j, kg/m3

σ surface tension, N/m

τWj wall shear stress of phase j, Pa

τi interfacial shear stress, Pa

39

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