NUMERICAL SIMULATION OF TWO-PHASE ANNULAR FLOW by

Joseph Michael Rodriguez

A Thesis Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

DOCTOR OF PHILOSOPHY

Major Subject: MECHANICAL ENGINEERING

Approved by the Examining Committee:

_________________________________________ Richard T. Lahey, Jr., Thesis Co-Adviser

_________________________________________ Kenneth E. Jansen, Thesis Co-Advisor

_________________________________________ Donald A. Drew, Member

_________________________________________ Michael Z. Podowski, Member

_________________________________________ Gretar Tryggvason, Member

Rensselaer Polytechnic Institute Troy, New York

July, 2009 (For Graduation August 2009)

CONTENTS

NUMERICAL SIMULATION OF TWO-PHASE ANNULAR FLOW............................ i

CONTENTS ...................................................................................................................... ii

LIST OF TABLES............................................................................................................. v

LIST OF FIGURES .......................................................................................................... vi

ACKNOWLEDGMENT ................................................................................................. xii

ABSTRACT ................................................................................................................... xiii

1. INTRODUCTION ....................................................................................................... 1

2. HISTORICAL REVIEW OF ANNULAR FLOW MODELING................................ 4

2.1 Annular Flow Description.................................................................................. 4

2.2 Multi-Field Two-Fluid Model............................................................................ 5

2.3 Overview of Closure Models ............................................................................. 7

2.3.1 Interfacial Shear ..................................................................................... 8

2.3.2 Droplet Entrainment............................................................................. 11

2.3.3 Droplet Size.......................................................................................... 13

2.3.4 Liquid Film Height............................................................................... 16

2.3.5 Lateral or Levitation Force................................................................... 18

3. OVERVIEW OF INTERFACE TRACKING METHODS....................................... 22

4. DESCRIPTION OF PHASTA-IC ............................................................................. 25

4.1 Finite Element Discretization of the Incompressible Navier-Stokes Equations.......................................................................................................................... 25

4.1.1 Governing Equations............................................................................ 25

4.1.2 Finite Element Formulation ................................................................. 26

4.2 Generalized Alpha Method .............................................................................. 29

4.3 Level Set Method ............................................................................................. 31

4.3.1 Governing Model ................................................................................. 31

4.3.2 Finite Element Formulation ................................................................. 33

ii

4.3.3 Volume Constraint on Level Set Redistance Step ............................... 34

4.4 Continuum Surface Tension (CST) Model ...................................................... 36

5. CANONICAL TEST PROBLEMS ........................................................................... 39

5.1 Rotating Semicircle.......................................................................................... 39

5.2 Analysis of 2D Dam Break .............................................................................. 42

5.3 Analysis of Solitary Wave ............................................................................... 46

6. PARALLEL MESH ADAPTATION........................................................................ 50

6.1 Code Description.............................................................................................. 50

6.2 Example Problem-A Convecting Sphere ......................................................... 52

7. SELECTION OF EXPERIMENTAL DATA FOR SIMULATION ......................... 55

7.1 Selection of Data Set........................................................................................ 55

7.2 Reducing the Experimental Data ..................................................................... 56

7.3 Estimating Minimum Turbulent Scales ........................................................... 57

7.4 Selecting the Test Run for the Simulation ....................................................... 58

8. SIMULATION OF TWO-PHASE ANNULAR FLOW ........................................... 62

8.1 Problem Description ........................................................................................ 62

8.1.1 Mesh Description ................................................................................. 63

8.1.2 Initial and Boundary Conditions .......................................................... 64

8.1.2.1 Effective-Viscosity Wall Function ........................................ 66

8.1.2.2 Verification of the Effective Viscosity Wall Function .......... 68

8.1.3 Simulation History ............................................................................... 68

8.2 Data Analysis Procedures ................................................................................ 69

8.2.1 Classification of Phase Fields .............................................................. 70

8.2.2 Evaluation of Data on Exit Plane ......................................................... 71

8.2.2.1 Description of Procedure ....................................................... 71

8.2.2.2 Computed Quantities ............................................................. 73

8.2.3 Evaluation of Wall Shear Stress........................................................... 75

iii

8.2.4 Evaluation of Total Local Shear Stress................................................ 76

8.2.5 Evaluation of Average Shear Stress Distribution................................. 78

8.3 Results and Discussion..................................................................................... 80

8.3.1 Void Fraction ....................................................................................... 87

8.3.2 Field Populations.................................................................................. 92

8.3.3 Field Mass Flow Rates ......................................................................... 94

8.3.4 Film Thickness ................................................................................... 102

8.3.5 Dispersed Field Size and Interfacial Area Density ............................ 107

8.3.5.1 Liquid Droplets .................................................................... 108

8.3.5.2 Steam Bubbles ..................................................................... 113

8.3.6 Wall Shear Stress ............................................................................... 118

8.3.7 Total Shear Stress & Interfacial Shear Stress .................................... 119

8.3.8 Film Thickness vs. Wall Shear........................................................... 126

9. CONCLUSIONS ..................................................................................................... 127

10. RECOMMENDATIONS FOR FUTURE WORK .................................................. 131

11. REFERENCES ........................................................................................................ 133

APPENDIX A: CLASSIFICATION OF PHASE FIELDS........................................... 140

iv

LIST OF TABLES

Table 1: RISO data for adiabatic steam/water tests in test section 20............................. 60

Table 2: Turbulent scales for adiabatic steam/water RISO data in test section 20.......... 61

Table 3: Simulation history ............................................................................................. 69

Table 4: Post processing methods used in evaluating the simulation data ...................... 70

Table 5: Experimentally determined mass flow rates through 30 degree section of

tube for RISO run#602. ....................................................................................... 97

Table 6: Averaged field mass flow rates and velocities. Values averaged over the

last 30 msec of the simulation. ............................................................................ 99

Table 7: Averaged field mass flow rates and velocities adjusted to simulate

numerical equivalent of the RISO test liquid film suction. Values averaged

over the last 30 msec of the simulation. ............................................................ 100

Table 8: Averaged field mass flow rates and velocities adjusted to simulate

numerical equivalent of the RISO test liquid film suction. Values averaged

over the last 30 msec of the simulation. ............................................................ 125

Table A1: Definition of Four Field values .................................................................... 142

v

LIST OF FIGURES

Figure 1: Two-phase annular flow..................................................................................... 4

Figure 2: Transition region of finite thickness 2 between fluids 1 and 2. ..................... 32 Figure 3: Schematic of two-dimensional level set test problem. Semicircle of

R=0.26 is placed in center of domain with a =2.0 prescribed velocity field. The domain was 0

Figure 12: Snap shots of the solitary wave calculation. The solitary wave is

considered generated 0.6 sec after the free fall of the initial profile. It is

designated with the t=0.0 marker. Grid: 200x120, wave speed=1 m/s. ............ 48

Figure 13: Snap shot of the velocity vectors of solitary wave at t=0.4s. The wave is

traveling to the right at a wave speed=1 m/s. ...................................................... 49

Figure 14: Solitary wave amplitude. (Grid: 200x120)..................................................... 49

Figure 15: Schematic of three-dimensional convecting sphere. A red water/green

water problem with surface tension. The computational domain was 0.5m

x 0.2m x 0.2m. ..................................................................................................... 53

Figure 16: Transport of a sphere through the domain using adaptive-mesh approach:

element size = 0.005m near interface and 0.02m elsewhere, mesh size ~

410,000 tetrahedrons. The black line indicates the interface. Mesh

displayed in upper half of planar cuts while the color spectrum in the lower

half indicates pressure (Pa). ................................................................................. 54

Figure 17: RISO test section 20 data plotted on the vertical flow regime map of

Hewitt and Roberts (1969) for thin tubes (1-3 cm) which was validated

against low pressure air/water and high pressure steam/water data. ................... 59

Figure 18: Schematic of the simulation domain. Length = 0.025m = 1/8 w for RISO Run # 602................................................................................................... 63

Figure 19: Boundary layer structure on wall. .................................................................. 64

Figure 20: Initial condition for RISO run #602 using plug flow velocity as initial

guess (ul=2.64 m/s, ug=4.99 m/s) and applying an imposed perturbation with

wavelength p = c............................................................................................... 65 Figure 21: Simulation solution at 63.9113 msec. (a) 3D interface of steam/water

interface colored by velocity magnitude, (b) phase solution at exit plane. ......... 73

Figure 22: Idealized annular flow.................................................................................... 79

Figure 23: 3D contours of steam/water interface at times in simulation. Contours

are colored by velocity magnitude. The dashed ovals indicate the location of

the large wave primarily responsible for droplet entrainment............................. 82

Figure 24: 3D contours of steam/water interface at times in simulation viewed from

exit plane. Contours are colored by velocity magnitude. ................................... 83

vii

Figure 25: Velocity fringe plots of center xy plane at times in simulation. Black line

depicts steam/water interface............................................................................... 84

Figure 26: 3D contours of steam/water interface at times in simulation. Contours

are colored by field (blue=steam bubble, red=liquid droplet, green=steam

core/ liquid film interface). The dashed ovals indicate the location of the

large wave primarily responsible for droplet entrainment................................... 85

Figure 27: 3D contours of steam/water interface at times in simulation viewed from

exit plane. Contours are colored by field (blue=steam bubble, red=liquid

droplet, green=steam core/liquid film interface). ................................................ 86

Figure 28: Simulation void fraction and associated error................................................ 89

Figure 29: Field area fractions at exit plane. ................................................................... 90

Figure 30: Disperse Field area fractions at exit plane: (a) dispersed liquid (DL), (b)

dispersed vapor (DV), (c) average over 30 msec. ............................................... 91

Figure 31: Maximum and minimum void fraction as a function of averaging time.

Oscillations in average void fraction curve are removed for averaging times

of 30 msec or greater. .......................................................................................... 92

Figure 32: Population of disperse fields in 30 degree sector during simulation: (a)

liquid droplets (DL), (b) steam bubbles (DV). .................................................... 93

Figure 33: Mass flow rates of continuous liquid (CL) and continuous vapor (CV)

fields as a function of simulation time................................................................. 97

Figure 34: Mass flow rate of disperse fields as a function of simulation time: (a)

liquid droplets (DL), (b) steam bubbles (DV). .................................................... 98

Figure 35: Averaged mass flow of fields as a function of simulation time: (a)

continuous fields (CL and CV), (b) disperse fields (DV and DL). Averages

performed over 30 msec. ..................................................................................... 99

Figure 36: Total mass flow rate as a function of simulation time. ................................ 100

Figure 37: Field axial velocity at exit plane as function of simulation time. ................ 101

Figure 38: Averaged axial velocity as a function of simulation time: (a) continuous

fields (CL and CV), (b) disperse fields (DV and DL). Averages performed

over 30 msec. ..................................................................................................... 102

viii

Figure 39: (a) Liquid film (CL) mass flow rate at exit plane as function of

simulation time, (b) Instantaneous and averaged liquid film thickness as

function of simulation time. Averages performed over 30 msec...................... 103

Figure 40: Continuous Liquid (CL) circumferentially-averaged area fraction at the

exit plane for four times at the end of the simulation. Each curve represents

an average over 15 msec.................................................................................... 104

Figure 41: Continuous Vapor (CV) circumferentially -averaged area fraction at the

exit plane for four times at the end of the simulation. Each curve represents

an average over 15 msec.................................................................................... 105

Figure 42: Disperse Liquid (DL) circumferentially -averaged area fraction at the exit

plane for four times at the end of the simulation. Each curve represents an

average over 15 msec......................................................................................... 105

Figure 43: Disperse Vapor (DV) circumferentially -averaged area fraction at the exit

plane for four times at the end of the simulation. Each curve represents an

average over 15 msec......................................................................................... 106

Figure 44: Comparison of predicted y(10%), y(50%), and y(90%) positions to the

experimentally measured values........................................................................ 106

Figure 45: Probability Distribution Functions (PDF) for (a) radial location of

droplets and (b) droplet size in terms of equivalent diameter, DEQ. Data

tallied from time steps 110,000 130,000 (time = 126.08 156.08 msec). ..... 109

Figure 46: Probability Distribution Functions (PDF) for radial location of droplets in

steam core for 12 ranges of droplet equivalent diameter, DEQ. The total

probability for each sums to 100 percent. Data tallied from time steps

110,000 130,000 (time = 126.08 156.08 msec). .......................................... 110

Figure 47: Probability Distribution Functions (PDF) for droplet size for 12 ranges of

radial locations. The total probability for each sums to 100 percent. Data

tallied from time steps 110,000 130,000 (time = 126.08 156.08 msec). ..... 111

Figure 48: Droplet Interfacial Area Density as a function of simulation history.

Simulation value ( iA =surface area/volume) are plotted with values assuming a spherical droplet ( iA =6/DEQ) where DEQ is the droplet equivalent diameter............................................................................................ 112

ix

Figure 49: Ratio of actual droplet interfacial area density to that assuming a

spherical droplet. Ratio is plotted against the simulation time. ........................ 112

Figure 50: Probability Distribution Functions (PDF) for (a) radial location of

bubbles and (b) bubble size in terms of equivalent diameter, DEQ. Data

tallied from time steps 110,000 130,000 (time = 126.08 156.08 msec). ..... 114

Figure 51: Probability Distribution Functions (PDF) for radial location of bubbles in

liquid film for 9 ranges of bubble equivalent diameter, DEQ. The total

probability for each sums to 100 percent. Data tallied from time steps

110,000 130,000 (time = 126.08 156.08 msec). .......................................... 115

Figure 52: Probability Distribution Functions (PDF) for bubble size for 9 ranges of

radial locations. The total probability for each sums to 100 percent. Data

tallied from time steps 110,000 130,000 (time = 126.08 156.08 msec). ..... 116

Figure 53: Bubble Interfacial Area Density as a function of simulation history.

Simulation value ( iA =surface area/volume) are plotted with values assuming a spherical bubble ( iA =6/DEQ) where DEQ is the bubble equivalent diameter............................................................................................ 117

Figure 54: Ratio of actual bubble interfacial area density to that assuming a

spherical bubble. Ratio is plotted against the simulation time. ........................ 117

Figure 55: Computed wall stress compared to experimental shear stress from RISO

run #602. Value from effective viscosity wall function computed from

Equation (138). Values for fully-developed assumption computed from

Equation (151) using data averaged over 30 msec. ........................................... 119

Figure 56: Total shear stress plotted against radial position for continuous liquid

(blue) and continuous vapor (red) fields over the time steps 65,000 95,000

where the axial pressure gradient was -1600 Pa/m. Each are plotted up to

the radial position of the 50 percent volume fraction. The average total shear

stress (green dashed) for steady, fully-developed annular flow is also shown

along with the steam volume fraction (black dotted). ....................................... 122

Figure 57: Total shear stress plotted against radial position for flow over the time

steps 65,000 95,000 where the axial pressure gradient was -1600 Pa/m.

The average total shear stress (green dashed) for steady, fully-developed

x

annular flow is also shown along with the steam volume fraction (black

dotted). ............................................................................................................... 123

Figure 58: Total shear stress plotted against radial position for continuous liquid

(blue) and continuous vapor (red) fields over the time steps 100,000

130,000 where the axial pressure gradient was -3300 Pa/m. Each are plotted

up to the radial position of the 50 percent volume fraction. The average total

shear stress (green dashed) for steady, fully-developed annular flow is also

shown along with the steam volume fraction (black dotted)............................. 124

Figure 59: Total shear stress plotted against radial position for flow over the time

steps 100,000 130,000 where the axial pressure gradient was -3300 Pa/m.

The average total shear stress (green dashed) for steady, fully-developed

annular flow is also shown along with the steam volume fraction (black

dotted). ............................................................................................................... 125

Figure 60: Illustration of how wall shear trends with film thickness @ exit plane: (a)

gradient of axial velocity w.r.t wall normal near the wall, (b) effective film

thickness. ........................................................................................................... 126

Figure 61: Computed wall shear stress compared to the experimental wall shear

stress for RISO run #602. The arrow shows the projected path of the

simulation assuming a linear increase in computed wall shear with

simulation time. ................................................................................................. 130

Figure A1: Illustration of walk-out procedure used to define a phasic glob. ................ 141

xi

ACKNOWLEDGMENT

I am grateful for the guidance and instruction from my advisors, Richard T. Lahey Jr.

and Kenneth E. Jansen. I have benefited from many hours discussing this research with

them and discussing how to proceed as the scope evolved over the years. This work

certainly would not have been possible without their involvement. Even as the time

passed, they remained committed to the research and inspired me to do the same. I

would like to also thank T. Darton Strayer who played a large role in me beginning this

research. He has provided continued technical input and encouragement from the start

and has been a great mentor and friend throughout the process. I also appreciate the

advice from Donald Drew who sat in on the countless meetings with us and provided

valuable insight. I would also like to thank the rest of my committee, Michael Podowski

and Gretar Tryggvason, who have taken time to evaluate my research and offer advice.

I am especially grateful to my wife, Mindy, and children, Ben and Anya, who have

patiently endured my journey. Never have they given anything but their full support and

I am forever thankful.

xii

ABSTRACT

A numerical simulation of a two-phase annular flow was performed using a three-

dimensional (3-D) stabilized finite element code, PHASTA-IC, with an implemented

level set method to capture the interface between the liquid and gas phases. The

problem simulated was run #602 of the experimental tests of Wurtz (1978), which was a

70 bar, adiabatic, steam/water annular flow in a 20mm I.D. tube having a total inlet mass

flux of 500 kg/s-m2 and an exit quality of 0.30. The mean experimental film thickness at

the exit of the tube was measured to be 0.94mm. The simulation modeled a 30 degree

segment of the tube of length 0.025m using a uniform tetrahedron mesh of edge length

size 0.00005m. The simulation, although not yet reaching equilibrium annular flow, was

able to capture the major mechanisms associated with annular flow. This includes

generation of instabilities on the interface between the steam core and liquid film,

formation of liquid ligaments that stretch into the steam core and shear off to form liquid

droplets, deposition of droplets back into the liquid film, the carry-under of steam

bubbles into the liquid film, and the development of large roll waves responsible for

most of these mechanisms. A classification tool was developed that interrogates the 3-D

solution and classified all entities in the domain into one of four fields: continuous liquid

(i.e., the liquid film), continuous vapor (i.e., the steam core), dispersed liquid (i.e., liquid

droplets in the steam core), and dispersed vapor (i.e., steam bubbles in the liquid film).

Various quantities, such as the mass flow rate, volume fraction, velocity, and interfacial

area density, were calculated for each field. In addition, methods were developed to

compute the total shear stress distribution, the interfacial shear stress, and the wall shear

stress. Comparisons were made to the experimental flow rates, shear stresses, and film

thickness.

xiii

1. INTRODUCTION

Multiphase flow occurs in many important industrial applications such as phase-change

heat exchangers, oil well pipelines, fossil-fired boilers and nuclear reactors. Many flow

regimes are possible, but the most common one in power production and utilization

technology is annular flow. This flow regime is quite complex and, as shown

schematically in Figure 1, it is characterized by a droplet laden gas core and a liquid film

on the conduit wall. The dynamics occurring on the wavy interface between the liquid

film and gas core are important to the process of liquid droplet entrainment into the gas

core, the lateral force responsible for keeping the liquid film on the conduit walls, and

the turbulence structures in the liquid and gas phases which effect the pressure drop.

Due to the difficulty in obtaining detailed local data in annular flow experiments,

researchers currently do not have a fundamental understanding of these interfacial

processes. With the advancement in modern computing capabilities, Direct Numerical

Simulation (DNS) can be used to help us understand the physical mechanisms in annular

flow (and other flow regimes) and generate numerical data to support the development

of physically-based closure laws to be used in advanced-generation 3-D, two-fluid

computational multiphase fluid dynamic (CMFD) models [Lahey (2009)].

As a result of the importance of annular two-phase flow, numerous researchers [Hewitt

and Hall-Taylor (1970), Henstock and Hanratty (1976), Tatterson et al. (1977), Kataoka

and Ishii (1983), Whalley (1987), Fore et al. (2000)] have developed phenomenological

models describing the physical processes of annular flow which were based more on

empiricism than detailed physical understanding. In mechanistically based two-fluid

computational multiphase fluid dynamics (CMFD) model formulations [Siebert et al.

(1995), Kumar et al. (2004), Lahey (2005)], which attempt to capture the dynamics of

two-phase flow, one uses closure laws based on the physical transport process at the

interface. Significantly, having accurate closure laws is important for both steady and,

especially, transient calculations.

Given that detailed experimental methods are not available to fully characterize the

wavy interface, one may use DNS data to support fundamental two-fluid model

development. That is, although DNS is not feasible as a practical tool for the design and

analysis of multiphase systems (due to the large computational expenses), Lakehal

1

(2004) and Lahey (2005) have shown how DNS may be used for closure model

development. This approach is consistent with a roadmap for basic research in

multiphase flow which was assembled by leading experts in the field [Hanratty et al.

(2003)], which supports the use of DNS (and improved experimental methods) to help

understand the microscopic mechanisms that affect the macroscopic behavior of

multiphase flows. The vast majority of DNS calculations done to date for two-phase

flows have focused on bubbly flow [Bunner and Tryggvason (1999), Tryggvason et al.

(2001), Nagrath et al. (2005)]. In contrast, very little has been done for annular flows

due to the computational costs associated with accurately capturing droplet entrainment

and the wave structure at the interface. Stability limits for stratified two-phase flows

have been investigated by Cao et al. (2004), Lakehal et al. (2003) and Fulgosi et al.

(2003), but these were done for relatively low Reynolds numbers such that wave

breaking and entrainment mechanisms were avoided. Nevertheless, some interesting

annular flow simulations have been performed previously. For example, Li and Renardy

(1999) simulated unsteady, axisymmetric oil/water annular flows using a Volume of

Fluid (VOF) method. Their computation was made at Reynolds numbers less than 10

and with a density ratio near unity. Consequently, wave breaking and droplet

entrainment processes did not occur. Also, using a level set method, Fukano and

Inatomi (2003) have simulated the formation of the liquid film around the tube wall in

horizontal air/water annular flow. They specifically investigated the source of the

levitation force that causes the liquid film to spread to the upper surface of the horizontal

tube. Although their simulation was able to capture a physical mechanism which could

explain the observed liquid film mobility, it was performed on a fairly coarse grid which

was incapable of resolving the droplets entrained in the gas core or any bubbles ingested

into the liquid film.

This investigation uses the PHASTA-IC code, with an implemented Level Set method,

to model two-phase annular flow. PHASTA-IC solves the discretized form of the

Incompressible Navier-Stokes (INS) equations in three dimensions using a stabilized

finite element method (FEM). A second-order accurate and stable generalized- time integrator [Jansen et al. (2000)] was applied to the INS equations and used to march the

solution in time [Whiting and Jansen (2001)]. PHASTA incorporates the level set

2

method of Sussman [Sussman et al. (1998), Sussman et al. (1999), Sussman and Fatemi

(1999)] and Sethian (1999) to resolve the interfaces in two-phase flows by modeling an

interface as the zeroth level set of a smooth function. The Continuum Surface Tension

(CST) model of Brackbill et al. (1992) was used to represent the local surface tension

force as a volume force applied over an interface of finite thickness.

Section 2 presents the multi-field two-fluid model and closure models relevant to two-

phase annular flow that may benefit from numerical data obtained using DNS. Section 3

briefly reviews the most common Interface Tracking Methods (ITM) in use today. The

theory and numerical implementation of the FEM and Level Set methods in PHASTA-

IC are given in Section 4. Section 5 presents canonical test problems used to verify

PHASTA-IC. An adaptive mesh refinement methodology of two-phase flows is

presented in Section 6 but, since it could not yet be supported by the Blue Gene

computer at RPI, it was not used in the simulation presented herein. Nevertheless, it

should be important for future simulations. The experimental data, to which the two-

phase annular flow simulation was compared, is described in Section 7. Finally, the

simulation and methods of reducing the data are discussed in Section 8 along with the

most salient results. Conclusions and recommendations for future work are presented in

Section 9 and 10, respectively.

3

2. HISTORICAL REVIEW OF ANNULAR FLOW MODELING

2.1 Annular Flow Description

In confined two-phase flows the annular flow regime exists when the liquid flow rate is

low and the gas flow rate is sufficiently high. It has been well investigated over the

years [Woodmansee and Hanratty (1969), Hewitt and Hall-Taylor (1970), Shedd and

Newell (2004)] and can occur in horizontal and vertical orientations. It is characterized,

Figure 1, by a liquid film on the wall containing entrained bubbles and a gas core

containing liquid droplets. In horizontal annular flow, the effect of gravity-induced

drainage increases the thickness of the liquid film on the bottom surface while reducing

it on the top surface. This effect diminishes with increasing gas velocity. Two types of

waves have been reported to exist on the interface: disturbance and ripple waves.

Disturbance waves are large amplitude roll waves that are responsible for the

entrainment of liquid droplets into the gas core. The base liquid film under these waves

is generally much higher than under ripple waves. Ripple waves are the low amplitude

surface waves which create interfacial roughness and are, thus, primarily responsible

for the pressure drop. Although these waves play a large role in the interfacial

dynamics, not enough is known about how they form and to what extent they influence

other aspects like droplet entrainment, gas and liquid turbulence, and interfacial shear.

Over the years this has lead to researchers constructing closure models based more on

empiricism than physics.

Figure 1: Two-phase annular flow

Liquid film

High Velocity Gas Core

Entrained gas bubbles in film

Liquid droplets in gas core

4

2.2 Multi-Field Two-Fluid Model

For development of mechanistically based two-fluid codes, the multi-field modeling

approach has proven to be effective in modeling the various flow regimes [Lahey

(2009)]. In this formulation, the flow is represented as a mixture of various

interpenetrating fields, for each of which the governing equations are solved. As the

number of fields increases the ability to model the flow increases; however, the burden

of developing the closure models describing the forces for each field and the interactions

with other fields limits the models to the minimum number of fields needed to capture

the dominant physics. Siebert et al. (1995) have shown that a four-field, two-phase

model is capable of capturing the flow regimes from bubbly to annular flow. Lahey and

Drew (2001) have also presented a four-field model which, without varying model

coefficients, can predict a wide variety of adiabatic and diabatic bubbly flows having

flow passages of various shapes. However, their model was not extended for annular

flows. A four-field implementation models the flow as the following idealized fields:

continuous liquid (CL), continuous vapor (CV), dispersed liquid (DL), and dispersed

vapor (DV). For each field the governing equations and turbulence models are solved

making it computationally costly to add additional fields, but the number of fields can be

reduced when modeling specific flow regimes. Trabold and Kumar (2000), and Faghri

and Sunden (2004) present a three-field approach for analyzing annular flow. They

neglected the role of gas droplets (DV) within the liquid film since there was no wall

heating. Their results compared favorably to local void fraction and droplet velocity

measurements for high void fraction (gas>0.75) adiabatic annular flows using refrigerant R-134a. The bubbly flow model presented by Kumar et al. (2004) models

only the disperse vapor and continuous liquid fields. They showed good agreement for

high pressure flow boiling in thin gap geometries when the void fraction was within the

bubbly flow regime.

Solving the Navier-Stokes equations for each phase requires a Direct Numerical

Simulation (DNS) of the flow to resolve the interfacial dynamics occurring on a

microscopic level. This may be feasible on a special basis for closure model

development but the vast amount of computational resources it requires precludes it

from being a tool for practical use [Lakehal (2004), Lahey (2005)]. Instead, the

5

governing equations of mass, momentum, and energy are averaged, effectively

averaging out the microscopic phenomena including capturing interfaces. The

information lost due to averaging is injected back into the solution via closure models,

the forms of which are dependent upon the physical phenomena being modeled. Of the

three averaging methods of space, time, and ensemble, Drew and Passman (1999), the

ensemble average was selected because it retains the spatial and temporal resolution of

the solution. An ensemble average is an average over all possible realization, , over a suitably large subset of realizations. The governing equations are ensemble averaged by

first multiplying each equation by the phase indicator function, Xk,

( )=01

, txX k (1)

and averaging over a large set of realizations. The average of the phase indicator

function is the void fraction for phase k, k. ( ) kk txX =, (2)

where x is the coordinate vector and t is time. All variables, f, are density-weighted

according to:

k

kk

k

kkk

XX

X

==

kk

kkk

k

kkkk

XfX

Xff ==

(3)

where is the density. Drew and Passman (1999) give the multi-field ensemble averaged equations for two-phase flow.

Continuity:

( ) ( ) jkjkkjkkjk mt +=+ jkv (4)

6

Momentum:

( ) ( )( )

jkjk

Re

jkjkj

vv

MMg

vvv

jkjk

wjkjkjkjkjkkjkjkjk

kjkkjk

m

pt

++++

=+

(5)

Energy:

( ) ( )( ) jkjkjkjkiijkjkTjkjkjkjkjk

kjkkjk

mAqDt

Dpqqq

t

jk

++++++

=+

D

jkjkjk hh v (6)

where the subscript jk represents field j of phase k, is the void fraction, is the density, is the interfacial mass transfer rate due to phase change, m is the mass source density of fluid from other fields, p is pressure, is the stress tensor, is the

Reynolds stress tensor, M and represent the interfacial and wall forces per unit

volume, respectively, is the volumetric heat generation rate, is the interfacial heat

flux, is the wall heat flux, is the so-called turbulent heat flux, is the interfacial

density, D is the dissipation function, and is the internal energy.

Re

wM

q iq q Tq iA

The underlined terms in the governing equations represent the interfacial transfer terms

that require closure models. These represent phenomena like interfacial exchange of

momentum through droplet entrainment and deposition, interfacial shear, and phase

change.

2.3 Overview of Closure Models

When the conservation equations are ensemble-averaged, the microscopic physics

associated with the interfaces and wall is lost. This includes such dynamic processes as

droplet entrainment into the vapor core, interfacial and wall shear, effect of the wavy

interface on the liquid film turbulence, and the droplet size distribution. Closure models

describing the relevant physics are typically used to recover the information lost during

averaging. Ideally, a closure model will be mechanistically based and accurately model

7

the physical phenomena. Due to the complexity of annular flow and the difficulty of

obtaining local information in the flow, most closure models to date are instead

correlations (using non-dimensional numbers) which are deemed to be relevant to the

physics of interest. While correlations are often numerically stable models, they are

suspect when used outside the range of data against which they were correlated.

Furthermore, most are correlated against equilibrium conditions and thus have limited

use in transient calculations.

Some of the closure models for annular flow are discussed below. They have been used

in a variety of computer codes to successfully predict data and so have been an asset to

date. However, their use is restricted by the data range over which they were correlated

and by the assumptions made in developing the models. Future fundamental codes will

require more mechanistically based models that adequately model the physics. It is with

these models that extrapolations beyond experimental data may be made. The

development of these improved models will require knowledge of local physics that may

be obtained through many simulations of annular flow conditions. This thesis discusses

the ability to perform one of these simulations but does not address formulation of the

improved closure models. However, for completeness, the forms of popular closure

models for annular flow are discussed below.

2.3.1 Interfacial Shear

The wavy interface on the liquid film creates a shear on the high-speed gas core more so

than had the gas core been traveling along a smooth wall. Recognizing this, Wallis

(1969) likened the wavy interface to a roughened wall and modified the single phase

friction factor, fsp, to compute the interfacial friction factor, fi. He used the non-

dimensional length scale of liquid film height, , to tube diameter, D, to obtain:

+=D

ff spi3001 . (7)

Noting the gas velocity to be more significant than the liquid film velocity, the

interfacial shear, i, is computed using the single-phase wall friction model. 2

21

ggii Uf = (8)

8

Although many authors have modified Wallis model over the years (e.g., to use relative

velocity rather than Ug), the basis form of Equation (7) has constituted the leading

models in use today. Henstock and Hanratty (1976) noted the difficulty of knowing the

film thickness height, , a priori, and developed a correlation for the film height based on the gas core Reynolds number, ReG, the liquid film Reynolds number, ReLF, and the

ratio of the liquid to gas densities and viscosities. The interfacial friction factor was

given by:

( )( )

),,Re,(Re

Flow Horizontal 8501

Flow Veritical14001

G

L

G

LGLF

spi

spi

FunctionF

FffFff

=+=+=

(9)

Noting that the tube diameter does not have as much influence on the friction factor as

given in previous models, Asali et al. (1985) used a friction length scale to scale the

single phase friction factor according to /lf instead of /D, where lf is given by:

5.0

=

g

i

gfl

. (10)

Ambrosini et al. (1991) refit Asalis correlation to account for a wide range of air/liquid

flows, flow rates, and pipe diameters. His modified interfacial friction factor is given

by:

+= +

l

gGgDspi mWeff

200Re8.131 6.02.0 . (11)

WeD, ReG, and mG+ are the Weber number based on tube diameter, gas Reynolds

number, and non-dimensional film thickness given by:

g

gG

g

gG

ggD

mUm

DUDUWe

*2Re === + , (12)

where Ug* is the friction velocity which is equal to (i/g)0.5. Additional modifications have been made by Brauner and Maron (1993) who added a memory effect to improve

calculations for transition between smooth and wavy stratified flow , Fukano and

Furukawa (1998) who tested the effects of varying viscosity on interfacial shear and

9

added a viscosity correction term, and Fore et al. (2000) who shifted the correlation to fit

an expanded database.

Ishii and Mishima (1984) departed from the linear relationship between the interfacial

and single phase friction factors by introducing a power law relationship between the

wall shear, w, and interfacial shear: m

wwi R

R

, (13)

where R and Rw are the local radius and tube radius, respectively. Similarly, Dobran

(1983) chose to use a power wall relationship to compute the effective momentum

diffusivity, eff, in the wavy region:

( )ntl

eff C ++ +=

11 , (14)

where l is the momentum diffusivity for liquid only, + is the average film thickness, t+ is the thickness of the liquid film to the wave troughs (.i.e. thickness of the base film),

and C1 and n are constants determined from the data. The underlying assumption is that

the base film can be modeled similar to single phase flow but the wavy top of the liquid

film should be a function of the wavy height and not the distance to the wall. The

different length scales will lead to a modified diffusivity. Dobran derived an expression

for t+ by correlating it to air/water data. The law-of-the-interface approach by Kumar and Edwards (1996) attempted to reduce the amount of correlation involved with

modeling the interfacial shear by using an analogous law-of-the-wall relation to directly

compute shear from the turbulent kinetic energy. Noting the single-phase shear, , is related to the turbulent kinetic energy, k, by:

kC = , (15) Kumar and Edwards (1996) assumed a similar relationship for the interfacial shear:

( )+=

=

ukC

T

uuT

gm

ilmi

5.025.0

, (16)

where the non-dimensional velocity, u+, is given by the log relationship similar to the

law-of-the-wall and the terms have been normalized against the interfacial velocity,

10

shear, and film thickness. ul and ui are the liquid and interfacial velocities. They were

unable to show agreement with data, citing issues with the droplet entrainment model

they used in their model.

2.3.2 Droplet Entrainment

Most researchers acknowledge liquid droplets are formed in the gas core via one of three

mechanisms; breakup of liquid bridges carried over from the transition to annular flow,

disintegration of larger droplets into smaller droplets, and entrainment from the liquid

film. The high speed photographic study of Woodmansee and Hanratty (1969) of

parallel air flow over water found the latter mechanism to be dominant due to a Kelvin-

Helmholtz instability forming on the back of the large disturbance waves. Small

wavelets were found to form on top of the disturbance waves, accelerate behind the

wave due to the lower pressure, be lifted by the high-speed gas core, and then shatter,

entraining droplets into the core. Considering the key forces acting on the wavelet along

the interface to be the drag and surface tension, Woodmansee and Hanratty selected the

ratio of these forces, the Weber number, to be the critical parameter governing the onset

of entrainment:

( )

2CUlWe ggc

= , (17)

where lc is a characteristic length of the wavelet, Ug the velocity of the gas core, C the

waves celerity, g the gas density, and the surface tension. Their experiment showed the critical air velocity for the onset of entrainment to increase with surface tension and

decrease with decreasing film thickness. Viscosity was not found to be of primary

importance.

Although some authors have done simplified force balancing and instability analyses to

generate templates for the droplet entrainment model, most are still correlated against

various data sets using various non-dimensional numbers, including We. Based on the

mechanistic model that entrainment occurs by the shearing of wavelets on disturbance

waves, Kataoka and Ishii (1983) and Ishii and Mishima (1984) developed models for

entrainment for three zones of the entrainment region of air/water data. The first zone is

the geometry dependent region in which entrainment can be very sensitive to the inlet

11

condition of the tube but is usually very small. They found this region to exist for z <

160DWe0.25Ref-0.5, where z is the axial coordinate along the tube and D is the tube

diameter. We and Ref are the Weber and liquid Reynolds numbers, respectively, and are

defined by:

=

g

gg DjWe

2

and l

llf

Dj

=Re , (18)

where jg and jl are the gas and liquid superficial velocities and l is the liquid viscosity. The last region is the equilibrium region where the entrainment and deposition rates are

equal and begins at:

5.025.0

Re440

f

WeDz = . (19) The entrainment fraction, E, defined as the ratio of the droplet volumetric flux, jfe, to the

total liquid volumetric flux, jl, is given by:

( )25.025.17 Re1025.7tanh fl

fe WexEjj

E == . (20)

The entrainment rate, , defined as the mass of the entrained droplets per unit time per unit interfacial area, is given by:

( ) 26.0925.07 Re106.6

=

l

gf

l

WexD

. (21)

The region between the equilibrium and geometry dependent regions is the developing

entrainment region where the entrainment rate was correlated to be:

( )

( ) ( ) 185.026.0925.07

225.075.19

1Re106.6

11Re1072.0

EWex

EEEWexD

l

gf

fl

+

=

. (22)

Dallman et al. (1984) correlated air/water data with different tube diameters to include

the effect of tube diameter on entrainment. They gave the following correlation for

entrainment fraction in the equilibrium region:

12

( )( )[ ]( )( )[ ] 5.1325.08

5.1325.08

2106.31

2106.31

glg

glg

L

LFC

UmDx

UmDxW

WE

+

=

, (23)

where WLFC is the critical liquid film flow rate below which entrainment does not occur,

WL is the liquid flow rate, and m is the average film thickness. Lopez de Bertodano et

al. (1995) performed separate effects tests using air/water and Freon as working fluids to

see the effects of surface tension and density ratio. Their correlation for entrainment rate

was a merger of those of Dallman et al. (1984) and Kataoka and Ishii (1983) and is given

by:

( )5.0

ReRe4

=

g

lgLFCLF

A

l

WekD

, (24)

where kA is a dimensionless atomization coefficient, ReLF is the liquid film Reynolds

number, and ReLFC is the critical film Reynolds number below which entrainment does

not occur. Holowach et al. (2002) developed a model, derived from stability analysis

and simplified force balances, which they felt would be better suited for transient

calculations. Their model for entrainment rate, using the symbol SE rather than , is given by:

( )( ) 5.03

,3

8/7

67.1

1Re0311.0

l

fgclwentr

g

lfilmE

UUVNS

= (25)

Refilm is the film Reynolds number, N is the viscosity number, is the wavelength of the interfacial wave, Ventr,w is the volume of liquid entrainment from the wave crest,

gcU is the average core velocity, and fU is the average film velocity. Although this

model is mechanistic in origin, Holowach et al. (2002) used correlations for the droplet

drag coefficient and interfacial friction factor in the derivation.

2.3.3 Droplet Size

The majority of the droplet size models are based on the assumption that the droplet size

is set during entrainment via a Kelvin-Helmholtz instability as seen by Woodmansee and

Hanratty (1969). Several variations of the model by Tatterson et al. (1977) have been

generated to correlate with an expanding experimental database. A smaller set of

authors, like Kocamustafaogullari et al. (1994), assume the droplet size may exceed the

13

maximum entrained droplet diameter during entrainment and that the final size is set by

droplet disintegration mechanisms in the gas core.

Tatterson et al. (1977) did a force balance about a liquid ligament prior to entrainment in

the core and used Kelvin-Helmholtz theory for inviscid flow over a small amplitude

wave to determine the pressure gradient as a function of wave number, k. Using friction

velocity, Ug* = (i/g)0.5 as the characteristic velocity scale and assuming k ~ 1/m for thin films, where m is the film height, the droplet diameter, d, was given by:

2/1*

=

mUmd gg . (26)

They found m and i using the correlations from Henstock and Hanratty (1976). Note there is no dependence on liquid film parameters. As suggested by data, the droplet

distribution could be modeled using an upper limit log normal distribution of the form:

( )( ) ( ) ( )

=2

max

2

max

max lnlnexp add

dddd

dddvd

, (27)

where v is the volume fraction of droplets having diameters less than d, dmax is the

maximum drop diameter, is a size distribution parameter, and a is a dimensionless parameter defined by:

vm

vm

ddda = max

, (28)

where dvm is the volume droplets median diameter. Droplets with d > dvm occupy 50%

of the volume. Evaluations of current air/water data showed and dmax/dvm are weakly dependent upon the flow parameters and may be assumed to be constant. Tatterson et al.

assumed the average value from the data: =0.72, dmax/dvm=2.90. They computed the mean volume diameter as:

2/122

21060.1

=

DfUx

Dd spggvm . (29)

Ambrosini et al. (1991) took the same approach at Tatterson et al. when developing a

correlation for a wider range of data that included varying liquids, gases, and pipe

diameters, but used their own correlations for m and i. They also included terms that

14

provided a density ratio correction, accounted for droplet collisions in the gas core, and

corrected the correlation for low core velocity data. Their correlation is given by:

+

=

Dgg

LE

l

g

gig WedUDG

mUfmd 0.9960.0exp0.22

32

83.05.0

232

. (30)

Note their correlation is for the Sauter mean diameter, d32, for which droplets having

diameters above d32 account for 68% of the droplet volume. GLE is the mass flux of the

entrained liquid and WeD is the Weber number defined in Equation (12).

Other relevant correlations are provided by Ueda (1979) and Kataoka et al. (1983).

Ueda assumed a gamma distribution for the droplet density function and, comparing

against data from air/liquid systems using water, glycerol, and aqueous alcohol

solutions, they developed the following correlation for the maximum droplet diameter: 34.025.1

3max 108.5

=

l

g

ggUxd

. (31)

Kataoka and Ishii (1983) based their correlation solely on air/water data and gave the

following expression for the mean volume diameter, dvm: 3/23/1

3/26/12 ReRe028.0

=

l

g

l

ggf

ggvm j

d

. (32)

Ref, Reg, and jg are the film Reynolds number, gas Reynolds number, and gas superficial

velocity, respectively. They found dmax = 3.13dvm which is similar to that of Tatterson et

al. (1977).

A different approach was taken by Kocamustafaogullari et al. (1994). They based their

model on the maximum droplet diameter being set by droplet disintegration in the gas

core and not by the entrainment process. A critical Weber number could be constructed

when doing a force balance between the surface tension force acting to maintain the

droplet and the disrupting shear placed on the droplet by the core flow. Two disrupting

forces were considered and a correlation for dmax derived from each. First, for flows

where the density of the disperse phase, d, is less than the continuous phase density, c, (i.e. bubbles in liquid medium), the disrupting force in primarily due to turbulent

fluctuations in the liquid film acting on the bubble interface. However, when d > c

15

(i.e., droplets in the gas core) the disrupting force results from the local relative motion

about the droplet interface. For this case, dmax is given by: 15/415/14

5/315/4*max Re

Re609.2

=

l

g

l

g

f

gmw WeCd

(33)

where dmax* = dmax/dh is the non-dimensional diameter, dh is the hydraulic diameter of

the flow channel, Wem is the Weber number based on the hydraulic diameter and gas

superficial velocity, and Cw is a coefficient defined by:

5.05.0

15/4

151N

34.351

151N25.0

=

=

g

N

NC

f

f

w

. (34)

Kocamustafaogullari et al. (1994) found dmax/dvm=2.93 and dmax/d32=4.01. A comparison

of the volume mean diameter models from Tatterson et al. (1977), Kataoka et al. (1983),

and Kocamustafaogullari et al. (1994) was done by Fore et al. (2002) against high

pressure nitrogen/water droplet data at system pressures on 3.4 and 17 bar. Only the

correlation from Kocamustafaogullari et al. (1994) reasonably predicted dvm for the

nitrogen data; the other correlations under-predicted the data. However, the coefficient

had to be increased to predict the correct Sauter mean diameters.

2.3.4 Liquid Film Height

The height of the liquid film, , has been primarily correlated against the liquid film Reynolds number, ReLF, with some authors including other terms to account for density

ratio, viscosity ratio, and buoyancy effects. One of the earliest models is by Kosky

(1971) who performed a force balance of the liquid film and, assuming the velocity

profile in the liquid film to be that for single phase turbulent flow, computed the average

film height. Since the turbulent profile used depended on the height of the film, Kosky

derived the following two film thickness equations:

16

)Re(high 25for Re0.0504

)Re (low 25for Re2

LF8/7

LF2/1

>=

and Reg is the Reynolds number based on gas velocity and tube diameter. Although they

found Koskys high ReLF model predicted their data, Fukano and Furukawa (1998)

correlated their air/water and air/glycerol solution data to include the effect of the

buoyancy using the Froude number, Fr = U/(g0.5 D). They correlated the mean film

thickness as:

{ }xFrD LOGO

19.025.0 Re34.0exp0594.0 = , (42) where FrGO, ReLO, and x are the Froude number for the gas core, liquid Reynolds

number, and the flow quality defined as:

Dg

jFr gGO = ,

l

llLO

Dj

=Re , ( )llgg gg jjj

x += (43)

and jl and jg are the liquid and gas superficial velocities, respectively.

2.3.5 Lateral or Levitation Force

Perhaps the biggest unknown associated with annular is the mechanism by which the

liquid film forms on the walls of the conduit, particularly on the upper surface in

horizontal annular flow. The four predominant theories that have been presented are:

Droplet entrainment and deposition Secondary gas flow Wave spreading Pumping action of disturbance waves

Each theory may be supported by experimental, and sometimes numerical, evidence.

Neglecting the secondary gas flow theory, the other theories agree the disturbance waves

play a leading roll in maintaining the liquid film, whether it be by generating liquid

droplets in the vapor core or causing the film at the bottom to be spread or pumped

upwards. Instantaneous measurements of film velocity and direction are required before

the physical mechanism may be understood. Given that the current experimental

techniques provide averaged (time and space) data, perhaps the best method is the use of

DNS. For completeness, each theory is discussed below.

18

Droplet Entrainment and Deposition:

Russell and Lamb (1965) hypothesized that for horizontal annular flow the liquid film on

the upper wall drains down the horizontal tubes wall but is continuously replenished by

impacting liquid droplets from the vapor core. Their experiments measured the salt

concentration at circumferential locations around the tube. Based on the salinity

measurements and the location of the salt injection point, the direction and velocity of

the circumferential flow could be inferred. They found the flow at the top surface to

flow downward on a time-averaged sense; however, there were fluctuations in which the

flow could flow upwards. They reasoned the bulk drainage of the liquid would drain the

liquid film unless droplets replenished the liquid supply. An effective circumferential

diffusivity was used to model the net exchange of liquid though drainage and droplet

supply.

Secondary Gas Flow:

The idea that secondary flows in the vapor core were responsible for pulling liquid up

the wall was proposed by Laurinat et al. (1985) and supported by Lin et al. (1985).

Secondary flows may be generated as a result of either flow traveling through a non-

circular cross section generated by a circumferentially varying film thickness or

circumferentially varying interfacial roughness. These secondary flows then shear the

liquid film and act to pull the liquid up the wall. Flores et al. (1995) used vorticity

meters to measure the presence of secondary flows in air-water horizontal annular flows.

At the low air flow rates used, the annular flow was near its inception point, minimizing

the importance of disturbance waves. The circumferential wall shear, s, on the liquid was proposed and based on the gas velocity, Vg, gas density, g, tube diameter, D, and surface roughness, :

( )[ ] ( )[ ] += 1433.3log1751047.6 25.1 DEVDE ggs . (44) No conclusions could be made of the importance of secondary flow at high flow rates

where disturbance waves play a larger role. Jayanti et al. (1990) evaluated the time-

dependent behavior of air/water annular flow based on measurements of circumferential

velocity and frequency. They found the film thickness on the upper surface correlated

with the frequency and strength of the disturbance waves. They also reasoned the

secondary flow to be too weak to produce the film thickness measured.

19

Wave Spreading:

Butterworth and Pulling (1972) proposed that the orientation and velocity of the

disturbance waves at the bottom of the tube result in a net flow of liquid toward the wall.

This results in the fluid spreading up the wall. Gravity waves will travel faster where the

film thickness is thicker the tube bottom. This variation in velocity results in the

disturbance waves having velocity components directed up the tube wall. This effect is

thought to be stronger as the gas velocity increases and disturbance waves become

stronger, in which case the film thickness becomes more uniform around the tube. This

was supported though the experimental work of Jayanti et al. (1990) who found the film

thickness correlated with the strength of the disturbance waves.

Pumping Action of Disturbance Waves:

Fukano and Ousaka (1989) present a model that attributes the levitation force to the

disturbance waves, but not based on the wave spreading theory of Butterworth and

Pulling (1972) but rather based on a pumping action setup by a pressure gradient

induced by the disturbance waves. The height and velocity of the pressure waves is

greatest at the bottom of the tube. Stagnation regions in the gas behind the disturbance

waves are, therefore, greater at the bottom than the top. Considering the film to be

sufficiently thin, and considering the pressure in the film to be that of the gas phase, a

negative pressure gradient is generated in the disturbance waves from the bottom to the

top. This pressure gradient is responsible for pumping the liquid to the top of the tube

when disturbance waves pass. The non-dimensional pressure in the disturbance wave

due to the gas stagnation was given by :

( ) ( )[ ]s

DWgDWgg

sxx

CuCuCpC

4

22

21

11

== , (45)

where g is the gas velocity, ug1 and ug2 are the gas velocities far upstream and just behind (stagnation region) the disturbance wave, CDW is the velocity of the disturbance

wave, s is the wall shear stress for single phase gas flow, and C1 is a constant. The experimental investigation of Sutharshan et al. (1995) supports this model and the wave

spreading theory when they optically measured the film velocity in horizontal

air/kerosene annular flows using a dye tracing method. They found the liquid film at the

top surface to be continuously draining down the tube wall except when disturbance

20

waves passed and the liquid moved upward. Secondary flow was deemed not be a major

cause for the film development because had it been a leading factor the liquid film would

have moved upwards the majority of the time. The pumping action theory was further

supported by the DNS results for horizontal annular flow by Fukano and Inatomi (2003)

who was able to simulate the development of disturbance waves and resulting film

development on the upper surface of the tube. They showed the film development was a

result of the pumping action set up by the pressure gradient in the disturbance wave.

Unfortunately, the simulation was performed on a grid too coarse to capture high

frequency instabilities and adequately capture droplet entrainment, thereby preventing

droplet entrainment from establishing the film. Also, they noted the film velocity was

highest near the wall instead of near the interface, which is contrary to the secondary

flow theory.

21

3. OVERVIEW OF INTERFACE TRACKING METHODS

The open literature has an abundance of Interface Tracking Methods (ITM) that track or

capture interfaces for multiphase problems. While each method has its own advantages,

the most prominent methods used are the Volume of Fluid (VOF), Front Tracking (FT),

and Level Set methods. Solving the Navier-Stokes equations for multiple phases has

proven to be very challenging so most computations deal with single or multiple bubble

dynamics and free surface flows. Very little has been done to characterize the interfacial

dynamics of annular flow liquid films in which wave breaking and droplet entrainment

occurs. Categorizing the various ITM is a tedious task but, fortunately, there are good

reviews provided by Lakehal et al. (2002) and Tryggvason et al. (2001). Using the

categorization of Tryggvason et al. most ITMs may be considered to be one of four

types: Front Capturing, Front Tracking, Boundary-Fitted grids, or Lagrangian.

The front capturing schemes solve the flow on a stationary grid and resolve the interface

using a marker function. The most popular front capturing methods are the Marker and

Cell (MAC), Volume of Fluid (VOF), and Level Set (LS) methods. The MAC method

places mass-less particles at the interface and convects them with the flow. Their

position is used to infer the interface location and topology. Perhaps more popular, the

VOF method [Hirt and Nichols (1981) and Rider and Kothe (1998)] advects a marker

function which is the volume fraction of liquid in the cell volume. The function has a

value of 1.0 when the cell volume contains all of phase A, a value of 0.0 when it

contains phase B, and has a steep gradient near the interface where it varies from 0.0 to

1.0. The interface is defined where the function has a value of 0.5. Because the

interface needs to be inferred from the discontinuous marker functions in the MAC and

VOF methods, accurate interface reconstruction is the liability of these methods,

especially with sharp interfaces. Initially, the VOF method used the algebraic Simple

Line Interface Construction (SLIC) technique in which a line was used to mark the

interface in each cell. Significant improvement has been made recently using the

Piecewise Linear Interface Construction (PLIC) method of Rider and Kothe (1998). The

major advantage of VOF is that it conserves mass, something that is more difficult to

accomplish with the Level Set method. For a more complete review of VOF methods to

date, the reader is referred to the work of Scardovelli and Zaleski (1999).

22

The Level Set method was first introduced by Osher and Sethian (1988) with

improvements by Sussman et al. (1998) and Sussman and Fatemi (1999). Nice

overviews of the methods are provided by Osher and Fedkiw (2001) and Sethian and

Smereka (2003). In contrast to VOF, the LS method uses a smooth and continuous

marker function, , termed the level set function, to advect the interface. The level set function is the signed distance from the nearest interface with the interface defined as the

zero level set (=0). The interface is advected through the domain using a simple advection equation and, since the level set function is spatially differentiable

everywhere, the method can easily handle merging interfaces and can compute interface

curvature. The ability to handle sharp, merging, and fragmenting interfaces is the LS

methods greatest characteristic. To handle the jump in physical properties across the

interface the interface is modeled with a finite thickness over which the fluid properties

smoothly vary. Accordingly, interfacial forces like surface tension are applied using a

Continuous Surface Force model like that proposed by Brackbill et al. (1992). The

smeared interface does give rise to numerical errors that corrupt the distance field which

lead Sussman and Fatemi (1999) to introduce an extra redistancing algorithm to help

preserve the level set field. Although improved, the LS method is tainted with not being

able to definitively conserve mass. Although mass correction methods [Lakehal et al.

(2002)] have been proposed for the standard LS method, variants on the LS method are

the recent trend. One variant couples the LS and VOF methods [Sussman and Puckett

(2000), Sussman (2003), Son (2005), and van der Pijl et al. (2005)] to merge the

interface resolving ability of LS with the mass conservation of VOF. The improved

mass conservation is achieved with a more complicated numerical method. Another

approach is the particle level set method of Enright et al. (2002) that proposes advecting

mass-less particles with the interface to use to check and correct the zero level set.

Front tracking methods are characterized by solving the flow on a stationary grid while

tracking the interface with a lower order grid. One of the more popular FT methods in

use is described in Unverdi and Tryggvason (1992) and Tryggvason et al. (2001). The

interface grid is comprised of points with elements formed between them. As the

interface deforms and stretches in the domain, the interface grid is restructured to ensure

adequate resolution. To maintain adequate resolution of the interface, points are

23

removed in regions where they have become crowded and added in regions where they

are spaced too far apart. This restructuring of the front can be complicated for arbitrary

topologies in 3D computations. While physical discontinuities in properties and forces

do exist at the interface, these must be transferred to the fixed grid. This is usually done

via a smoothing operation which effectively transforms the interface forces in to volume

forces to be applied to the fixed grid. This is similar to the way the LS method uses the

CSF to apply surface tension over a finite thick interface.

Boundary fitted methods [Ryskin and Leal (1984) and Fulgosi et al. (2003)] are

probably the most rigorous methods but their application is restricted to simple

interfaces. The Navier-Stokes equations are first solved for each fluid separately over

their respective subdomain. They are then coupled at the interface via jump conditions

for continuity and momentum with the interface motion computed by solving an

advection equation for the interface height. This method cannot be extended to flows

with strong topological changes that may produce entrainment. For this reason it is not

applicable for modeling annular flow.

With Lagrangian methods [Hu et al. (2001) and Johnson and Tezduyar (1997)] the grid

follows the interface. In explicit Lagrangian schemes, the flow field is first solved using

the particle velocities and used to compute the forces acting on the particles. The

particles velocities are then computed and used to update the particle positions. The

mesh movement scheme used is not trivial with additional complexity added when

modeling colliding particles. The ability to properly follow breaking waves is suspect

and so Lagrangian methods are not well suited for annular flow simulations.

24

4. DESCRIPTION OF PHASTA-IC

PHASTA-IC solves the Incompressible Navier-Stokes (INS) equations in three

dimensions using a stabilized finite element method. The second-order accurate and

stable generalized- time integrator of Jansen et al. (2000) as applied to the INS equations [Whiting (1999)] is used to march the solution in time. PHASTA-IC

incorporates the Level Set method [Sussman et al. (1998), Sussman et al. (1999),

Sussman and Fatemi (1999), and Sethian (1999)] to resolve interfaces in two-phase

flows by modeling the interface as the zero level set of a smooth function. The

Continuum Surface Tension (CST) model of Brackbill et al. (1992) is used to represent

the local surface tension force as a volume force applied over an interface of a finite

thickness.

4.1 Finite Element Discretization of the Incompressible Navier-Stokes Equations

4.1.1 Governing Equations

The spatial and temporal discretization of the INS equations within PHASTA has been

described in Whiting (1999) and Nagrath (2004). This section follows their discussion.

The strong form of the INS equations is given by:

continuity: 0, =iiu (46) momentum: ijijijijti fpuuu , ,,, ++=+ (47)

where is density which is constant, ui is ith component of velocity, is pressure, p ij is the stress tensor, and represents body forces along the iif

th coordinate. For the

incompressible flow of a Newtonian fluid, the stress tensor is related to the strain rate

tensor, Sij, as:

( )ijjiijij uuS ,,2 +== . (48)Using the Continuum Surface Tension (CST) model of Brackbill et al. (1992), the

surface tension force is computed as a local interfacial force density, which is included

in . if

25

4.1.2 Finite Element Formulation

The INS equations apply over a spatial domain defined by NR where N equals three for our case. This domain is composed of the interior, , and the boundary, . Note the boundary is further subdivided into regions with natural boundary conditions,

h, and regions with essential boundary conditions, g, such that gh = . The finite element formulation is based on the so-called weak form of Equations (46) and

(47). To derive the weak form, continuous function spaces of trial solutions, S, and

weighting functions, W, are defined with square-integrable values and derivatives (H-1

functions) on . The domain is discretized into nel finite elements defining the element domain, e . Consequently, finite-dimensional approximations, or subspaces, of S and W, specified as Sh and Wh, respectively, are defined as:

( ) ( ) [ ] ( ) ( ){ }1 g, , 0, , , , o e

N Nkh kx

t H t T P t= = S u u u u g n (49)( ) ( ) [ ] ( ) ( ){ }g1 on ,,,,0,, == 0wwwwW tPTtHt NkxNkh e (50)

( ) ( ) [ ] ( ){ }NkxNkh PpTtHtppP e = ,,0,, 1 (51)where h represents the characteristic length of the parameterized domain, g is an

approximation of the prescribed boundary condition in the discretized domain, and

Pk( e ) is the piecewise polynomial space defined on the element e for order k 1. Note the velocity and pressure variables use the same polynomial space, which is

enabled through the stabilized formulation used in PHASTA-IC.

The so-called weak form is generated by dotting Equations (46) and (47) on the left by

the weight functions defined above and integrating over the domain. Following the

stabilized formulation of Taylor et al. (1998) the residuals of the continuity and

momentum equations are added together. The Galerkin finite element formulation is

given by finding , such that: khkh p and P Su

26

{ } { }[ ]( )

+++++==

dnpwnqu

dpwfuuuwuqpuqwB

puqwB

jijijiii

ijijjiijijtiiiiiiG

iiG

h

),;,(

0),;,(

,,,,

(52)

for all . khkh q and P Ww

The pressure, viscous stress, and continuity terms are integrated by parts eliminating all

second order derivatives. The boundary integral term in Equation (52) is a by-product of

this and only exists on h. Stabilization terms are added to the standard Galerkin method to correct known stabilization problems with advection dominated problems. The

resulting stabilized formulation used in PHASTA-IC is given by

finding , such that the residual, B, becomes: khkh p and P Su

0ion termsstabilizat),;,(),;,( >

introduced as a consequence of the momentum stabilization in the continuity equation.

The second term on this line was introduced to stabilize this new advective term. The

stabilization parameters, described in detail by Whiting and Jansen (2001), for

continuity, C, and momentum, M are given by:

ijijjiji

M

ggcuguct

cC

2322

1

++= and

(57)

( ))1 ijMC gtrace = . (58)The stabilization term for the conservation terms, ~ , is given by:

jiji uguc

C=

2

~ . (59)

The coefficients C, c1, c2, and c3 are defined based on the one-dimensional linear

advection-diffusion equation using a linear finite element basis. gij is the covariant

metric tensor related to mapping for global to parametric coordinates and is defined as gij

= k,ik,j where is the parametric coordinate. The weight functions and solution variables in Equation (54) are expanded in terms of

the finite element basis. The spatial integrals are evaluated using Gauss quadrature.

Since the weight functions are arbitrary, this results in a system of first-order, non-linear

ordinary differential equations that may be written in the form:

( ) s, n1A ,0,, "==puuR tiiA (60)where RA represents the residual error, A represents the values of the arbitrary weight

functions, and ui, ui,t, and p are the solution vectors in the discrete space. The

generalized- time integrator, discussed below, is then used to convert equation (60) to a system of non-linear algebraic equations. Newtons method is used to linearize the

problem about and . The system of equations may now be written as: 1, +ntu1, +ntp

=

+

+

C

M

n

n

R

R

p

u

CD

GK

1

1

, (61)

28

where RC and RM represent the portions of the residual, R, from the continuity and

momentum equation, respectively, and represents a modified (change in

pressure acceleration) to allow using Backward Euler for pressure while maintaining D =

-G

1 + np 1+ np

T. K, G, D, and C are tangent matrices of the residual vectors with respect to the state

variables and . Approximate forms of these tangent matrices are used to

achieve better convergence. As such, they are defined as:

1, +ntu1, +ntp

1111 ++++

n

C

n

C

n

M

n

M

pRC

uRD

pRG

uRK . (62)

The reader may refer to Whiting (1999) and Jansen et al. (2000) for a more complete

discussion of using the generalized- method to solve the stabilized finite element formulation of the incompressible Navier-Stokes.

4.2 Generalized Alpha Method

PHASTA-IC uses the Generalized- method Jansen et al. (2000) to integrate the spatially discretized Navier-Stokes equations from time step n to n+1. Whiting (1999)

presents its application to the incompressible Navier-Stokes equations. This is a

predictor-multi-corrector algorithm in which the solution is related to its time derivative

by:

( )nnnnn yytytyy ++= ++ 11 , (63)where is the solution vector, is its time derivative, y y t is the time step, is a weighting parameter, n denotes the previous time step, and n+1 denotes the current time

step being solved. The spatially discretized Navier Stokes equations may be expressed

as:

0),,( =pyyR , (64)where R is the residual of the continuity and momentum equations. Given a solution at

time step n, the generalized- method first makes a prediction of the solution and its time derivative at n+1. A series of correction passes are then made to improve this

initial guess. The prediction used in PHASTA-IC assumes the solution at n+1 matches

the previous solution at n:

29

nin

nin

yy

yy

10

1

01

=

==+

=+

, (65)

where i is the iteration on correction passes. For each correction pass the solution is

computed at an intermediate time. The solution is computed at a time n+f and the time derivative is computed at a time n+m and these are used to compute the residual from Equation (64). The intermediate solutions are defined by:

( )( )ninfnin

ninmn

in

yyyy

yyyy

f

m

+=+=

+

++

+

++

11

1

11

1

(66)

A Newton type linearization of the residual with respect to the time derivative of the

solution at n+1 yields the following equation:

iini

n

in

in

i

Ryy

y

yR f

f

=

++

+

+1

1

. (67)

Equation (67) represents a matrix problem in which the change in the solutions time

derivative, , at the iiny 1+ th iteration is computed. The solution at n+1 is then corrected according to:

in

in

in

in

in

in

ytyy

yyy

1111

1111

++++

++++

+=+=

(68)

As many correction passes from Equations (66) to (68) are performed to find the

solution at time n+1.

To obtain second order accuracy and satisfy stability, the three parameters, , f, and m are related in the following manner:

parameter defineduser

11 ,

13

21 ,

21

=+=

+=+=

fmfm (69)

Note that setting = f = m = 1 results in a Backward Euler solution.

30

4.3 Level Set Method

4.3.1 Governing Model

The level set method of Sussman [Sussman et al. (1998), Sussman and Fatemi (1999),

Sussman et al. (1999)] and Sethian (1999) involves modeling the interface as the zero

level set of a smooth function, , where represents the signed distance from the interface. Hence, the interface is defined by = 0. The scalar , the so-called first scalar, is convected within a moving fluid according to,

0=+= u

tDtD (70)

where u is the flow velocity vector. Phase-1, typically the liquid phase, is indicated by a

positive level set, > 0, and phase-2 by a negative level set, < 0. Evaluating the jump in physical properties using a step change across the interface leads to poor

computational results. Therefore, the properties near an interface are defined using a

smoothed Heaviside kernel function, H, given by:

( )

>

Fluid 1

Fsv 1, p1

Figure 2: Transition region of finite thickness 2 between fluids 1 and 2.

The fluid properties are thus defined as:

( ) ( ) ( )( )

( ) ( ) ( )( )

HH

HH

+=

+=

1

1

1

2

1

2

(72)

Although the solution may be reasonably good in the immediate vicinity of the interface,

the distance field may not be correct throughout the domain due to var